text:
## Phase Portrait Nonlinear System

1) Find all equilibrium points by solving the system 2) Let a standard software (e. for the analysis of nonlinear systems; to introduce controller design methods for nonlinear systems. Specific topics include maps and flows in one and two dimensions, phase portraits, bifurcations, chaos, and fractals. Phase Portraits and Time Plots for Cases A (pplane6) Saddle Ex. 1), which was diagnosed using a set of four features extracted from the phase plane trajectory of the system to characterize the nonlinear response in the periodic regime. Local Phase Portrait of Nonlinear Systems Near Equilibria. “Proof”: Consider trajectory sufficiently close to origin time reversal symmetry. Save the phase portraits to submit on Gradescope. Keywords: nonlinear dynamics, chaos, electrical circuits. Biological Models: Predator-prey models, Competition models, Survival of one species, Co-existence, Alligators, doomsday and extinction. • As much as possible, piece the phase portraits of the linearized systems together to get an approximate phase portrait of the full non-linear system. A: The origins and evolution of predator-prey theory, Ecology 73, 1530-1535 (1992). Consider the following phase portraits of two two-dimensional linear dynamic sys-tem What can you say about the real parts of the two eigenvalues for both systems? What creates the di erence between the two phase portraits? Is the equilibrium point in phase portrait (b) an attractor? Exercise 3 This is exercise 3. Often, mathematical models of real-world phenomena are formulated in terms of systems of nonlinear differential equations, which can be difficult to solve explicitly. This diagram clearly illustrates for what values of r, the system exhibits chaotic and non-chaotic behavior. See phase portrait below. 1 In each problemﬁnd the critical points and the corresponding linear system. Albu-Schaffer. The author starts off with an introduction to nonlinear systems, then moves on to phase portraits for 2-D systems, before moving on to advanced concepts of stability theory and feedback linearization. Stable and unstable manifolds of equilibrium points and periodic orbits are important objects in phase portraits. In this paper, based on the classic Chua's circuit, a charge-controlled memristor is introduced to design a novel four-dimensional chaotic system. Introduction to systems of differential equations 2. (reductor and multipliers). While nonlinear systems of-ten require complex idiosyncractic treatments, phase potraitshaveevolved as apowerfultool forglobal anal-ysis ofthem. The system lives in a state space or phase. Weak non-linear oscillators and. Existence, uniqueness, and strong topological consequences for two-dimensions. For a much more sophisticated phase plane plotter, see the MATLAB plotter written by John C. Simmons, Differential Equations with Applications and Historical Notes, New York: McGraw-Hill, 1991. We also observe that the two xed points are progressively pushed apart in the amplitude direction. For a general 2 × 2 matrix A, the phase portrait will be equivalent to one of the four cases above, obtained by a linear transformation of coordinates (similarity transformation). Fixed points are stagnation points of the ﬂow. The three examples will all be predator-prey models. 1 Concepts of Phase Plane Analysis 18 2. Consider the following phase portraits of two two-dimensional linear dynamic sys-tem What can you say about the real parts of the two eigenvalues for both systems? What creates the di erence between the two phase portraits? Is the equilibrium point in phase portrait (b) an attractor? Exercise 3 This is exercise 3. Click on the button corresponding to your preferred computer algebra system (CAS). 3 Symmetry in Phase Plane Portraits 22 2. Thus, a system has a limit cycle if and only if it has an isolated, closed. Numerical Construction of Phase Portraits. Two integral constraints on the amplitude and phase variation of the oscillations of an autonomous multi-degree of freedom system were obtained. 2 Draw the phase portraits of the following systems, using isoclines (a) 8+8+0. Generally, the nonlinear time series is analyzed by its phase space portrait. The book is very readable even though it has a lot of jargon (read heavy mathematics). A phase portrait is a graphical tool to visualize long term. Phase portrait generator. 5 0xy (7) which itself is a dynamical equation, the phase portrait is a trajectory along the switching line σ = 0. By plotting phase portrait on the computer, show that the system undergoes a Hopf bifurcation at 휇 = 0. shown to produce sharp phase portraits in long-term simulations, see e. Note: If you want a more traditional treatment of phase portraits, I recommend exploring Nonlinear Dynamics and Chaos by Strogatz. On this page I explain how to use Matlab to draw phase portraits for the the two linear systems. This video deals with. Existence, uniqueness, and strong topological consequences for two-dimensions. Simmons, Differential Equations with Applications and Historical Notes, New York: McGraw-Hill, 1991. The y nullcline is given by 3 4 1 4 3 y 2 3 x y = 0 (12) which gives the lines y = 0 or y = 3 4. , non-linear) 2 × 2 autonomous system discussed at the beginning of this chapter, in sections 1 and 2: x = f (x, y); (1) y = g(x, y). The nonlinear system's phase portrait near the fixed point is topologically unchanged due to small perturbations, and its dynamics are structurally stable or robust. Phase plane analysis for linear systems. 50= 1 (c) 8+82+0. m: A demonstration that plots the linearized phase portraits and the full phase plane. The nonlinear gyroscope model, which is employed in aerospace engineering [24], generally exhibits chaotic behavior. of problems that are described by nonlinear diﬀerential equations. The complex dynamics of the novel chaotic system such as equilibrium points, stability, dissipation, bifurcation diagrams, Lyapunov exponent spectra and phase portraits are investigated. EECS 222 Nonlinear Systems: Analysis, Stability and Control Shankar Sastry 299 Cory Hall Tu-Th 11-12:30 pm. The book is very readable even though it has a lot of jargon (read heavy mathematics). Phase portraits of nonlinear systems: predator-prey, van der Pol (MATLAB examples). Nonlinear Models and Nonlinear Phenomena. However, these behaviors are not properly depicted in phase portraits when dealing with sys-tems that could be described as rotating systems,. The department offers project courses where you may choose/propose a project on topics related to Nonlinear Dynamical Systems. ) Lecture, three hours; discussion, one hour. A two-state phase portrait approach has been used to analyse vehicle dynamics and provides an illustrative view of the state trajectories at constant speed. Analyze the stability and its margins. by graphing and the use of phase portraits; D. First, let us look at the phase space portraits for a range of phase advances from 0:2 2ˇto 0:5 2ˇ. The phase portraits is able to perfectly capture all of the nonlinear trajectories and display them in a way that would be otherwise difficult. The phase portrait behavior of a system of ODEs can be determined by the eigenvalues or the trace and determinant (trace = λ 1 + λ 2, determinant = λ 1 x λ 2) of the system. In previous work, it was shown that bang-bang trajectories with low values of the energy integral are optimal for arbitrarily large times. The equation governing the dynamics of the nonlinear gyro, enriched with linear and nonlinear smoothening terms [24], is given by x_1 = x2. The author starts off with an introduction to nonlinear systems, then moves on to phase portraits for 2-D systems, before moving on to advanced concepts of stability theory and feedback linearization. This paper extends the phase portrait to three states to represent the nonlinear vehicle dynamics with steering and longitudinal tyre force inputs and consideration of the longitudinal. (1) There is one equilibrium solution of this system – ﬁnd it! (2) Linearize the system near this equilibrium, and draw the phase portrait of the linearized system. This is a second order systemwhich is autonomous (time does not appear explicitly). from second-order equation to first-order system; what is a phase portrait; direction field of a first-order system; graphing in the xy- tx- and ty-planes; vector notation for a first-order system; semesters > spring 2020 > mth264 > resources > video > linear systems: basics Video | Linear Systems: Basics. Recall the basic setup for an autonomous system of two DEs: dx dt = f(x,y) dy dt = g(x,y) To sketch the phase plane of such a system, at each point (x0,y0)in the xy-plane, we draw a vector starting at (x0,y0) in the direction f(x0,y0)i+g(x0,y0)j. Damped Pendulum. Nonlinear Systems and Stability Autonomous systems and critical points Stability and phase plane analysis of almost linear systems Linearized stability analysis and plotting vector fields using a MSS Numerical solutions and phase portraits of nonlinear systems using a MSS Models and applications: TEXT: Text(s) typically used in this course. The classic Van der Pol nonlinear oscillator is provided as an example. 3 in Third and 3-42 Fourth Quadrant. We illustrate all these cases in the examples below. • Understand the linearized models using the “eigen-techniques” you learned earlier. (b) This plot includes the solutions (sometimes called streamlines) from different initial conditions, with the vector ﬁeld superimposed. The dynamical equation and the state equation of the system are established. We define the equilibrium solution/point for a homogeneous system of differential equations and how phase portraits can be used to determine the stability of the equilibrium solution. These states can also be correlated with velocity spectral behaviors. A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. (Hint: Use polar coordinates. The low intensity xed point appears on the phase portraits. 26 Phase Portrait for sand Y1 Magnitudes of 9. Now consider the nonlinear di erential equation = 1 2sin (6) Determine the equilibria of this system and their stability type. 50= 1 (c) 8+82+0. Its usage is also observed heavily in smart brakes systems of current automotive vehicles. 1 Phase portraits 72 3. warn(warning_msg, ODEintWarning). Is there a way for plotting phase portraits and vector fields for autonomous system of delay differential equations in. This video deals with. The dynamical equation and state equation of the system are established. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Cambridge: Westview Press, 2000. Phase Portraits of Nonlinear Systems Consider a , possibly nonlinear, autonomous system , (autonomous means that the independent variable , thought of as representing time, does not occur on the right sides of the equations). Damped Pendulum. One- and two- dimensional flows. to sketch the phase portrait. Stable and unstable manifolds of equilibrium points and periodic orbits are important objects in phase portraits. In this paper, based on the classic Chua's circuit, a charge-controlled memristor is introduced to design a novel four-dimensional chaotic system. I have a set of three differential equations and I want to make a phase portrait of them. For a chaotic system, there will be many distinct loops in a phase portrait, showing that the system is aperiodic and does not approach a stable. Quiver function is being used for phase portrait plots obtained using ode. 1 Solution curves in the phase plane of the Lotka-Volterra predator-prey model102 6. - A ”limit cycle” is a periodic orbit that trajectories approach. 3 Determining Time from Phase Portraits 29 2. 3 Symmetry in Phase Plane Portraits 22 2. Derive the dynamics of a linear and nonlinear systems. Fig 1: 3-D phase portrait of the Rossler attractor Integration Solver The Runge-Kutta method for a system of ordinary differential equations is explained here. The dynamical variables of the system, in this case the angular position and velocity , are the coordinates defining the system's phase space. And it turns out, with this omega two this was the separatrix case, but that was the intermediate axis case. Sketching an accurate phase portrait for a non-linear system of DEs is time consuming but the series of 3 videos will help with shortening that time with added understanding. Phase Plane Portraits of Almost Linear Systems Interesting and complicated phase portraits often result from simple nonlinear perturbations of linear systems. 2 Bifurcation set and phase portraits of the Hamiltonian system (5). Ask Question Asked 4 years, 7 months ago. Solving 2x2 homogeneous linear systems of differential equations 3. According to Takens, almost all d-dimensional sub-manifolds could be embedded in a (m=2d+1) dimensional space. -----, Phase portraits of non degenerate quadratic systems with finite multiplicity one, Nonlinear Anal. These programs provide animated phase portraits in dimension two and three, i. The tem-poral response of a system need not be the. 2 : Linear analysis of nonlinear pendulum : Mechanical systems model for a pendulum. (a) This plot shows the vector ﬁeld for a planar dynamical system. , another nonlinear system x_1 = 1 x3 1 x_2 = x1 x22 equilibrium points are described by x1 = 1 and x2 = 1 note: the equilibrium points of a nonlinear system can be nite (2 in the previous examples, but any other number is possible, including zero) or in nite, and they can be isolated points in state space Oriolo: Stability Theory for. For more information on phase portraits and types of fixed points for linear systems of ODEs, see, for example: S. Phase Plane Analysis 17 2. Many nonlinear dynamic systems have a rotating behavior where an angle defining its state may extend to more than 360∘. Weak non-linear oscillators and. Its usage is also observed heavily in smart brakes systems of current automotive vehicles. Ott, and A. This suggests that the only. , regularly timed speech with a metronome). By plotting phase portrait on the computer, show that the system undergoes a Hopf bifurcation at 휇 = 0. 6: Phase portraits on the (one-dimensional) centr emanifoldandthebifurcation diagram. First, we cover stability definitions of nonlinear dynamical systems, covering the difference between local and global stability. We define the equilibrium solution/point for a homogeneous system of differential equations and how phase portraits can be used to determine the stability of the equilibrium solution. 1 (Saddle) Consider the system x˙ 1 = −x 1 −3x 2 x˙ 2 = 2x 2, x. in weakly-nonlinear systems was investigated. 1 Phase Portraits 18 2. The complex dynamics of the novel chaotic system such as equilibrium points, stability, dissipation, bifurcation diagrams, Lyapunov exponent spectra and phase portraits are investigated. It is not restricted to small or smooth nonlinearities and applies equally well to strong and hard nonlinearities. warn(warning_msg, ODEintWarning). Limit Cycles. (1) There is one equilibrium solution of this system – ﬁnd it! (2) Linearize the system near this equilibrium, and draw the phase portrait of the linearized system. The aim of this section is to present programs allowing to high- light the slow-fast evolution of the solutions of nonlinear and chaotic dynamical systems such as: Van der Pol, Chua and Lorenz models. x′= x−y, y′= x2 +y2 −1 2. Four possible phase portraits for this system are shown along the right side of the page. This course introduces the main topics of low-dimensional nonlinear systems, with applications to a wide variety of disciplines, including physics, engineering, mathematics, chemistry, and biology. Many nonlinear dynamic systems have a rotating behavior where an angle defining its state may extend to more than 360∘. doc Author: tien Created Date: 11/15/2002 4:16:10 AM. By varying the initial conditions of the system, it is found. The type of phase portrait of a homogeneous linear autonomous system -- a companion system for example -- depends on the matrix coefficients via the eigenvalues or equivalently via the trace and determinant. Phase portrait. In paper [12], the vibration of a typical two degree of freedom nonlinear system with repeated linearized natural frequencies was investigated systematically. We discovered the system’s rich behavior such as chaos through phase portraits, bifurcation diagrams, Lyapunov exponents, and entropy. The course revises some of the standard phase portrait methods encountered in the Dynamical Systems course in part II and extends these ideas, discussing in some detailed centres, via the use of Lyapunov functions, limit cycles and global phase portraits. 1 Concepts of Phase Plane Analysis 18 2. Keywords: nonlinear dynamics, chaos, electrical circuits. A phase portrait is mapped by homeomorphism, a continuous function with a continuous inverse. Linear approximation of autonomous systems 6. Let us consider the below block diagram of a non linear system, where G 1 (s) and G 2 (s) represent the linear element and N represent the non linear element. Existence, uniqueness, and strong topological consequences for two-dimensions. • As much as possible, piece the phase portraits of the linearized systems together to get an approximate phase portrait of the full non-linear system. The complex dynamics of the novel chaotic system such as equilibrium points, stability, dissipation, bifurcation diagrams, Lyapunov exponent spectra and phase portraits are investigated. Damped Pendulum. Phase Plane Analysis 17 2. If the stable manifold is of higher dimension, then y 1 =h 1(x,µ ),y 2 =h 2(x,µ )and. There are lots of practical systems which can be approximated by second-order systems, and apply phase plane analysis. (e) Draw the phase plane portrait of the nonlinear system via v and h nullclines. The resonance effects are most pronounced where both the. 4 Phase Portraits and Bifurcations. A numerically generated phase-portrait of the non-linear system Zoomed in near (0,0) Zoomed in near (2,1) The critical point at (2,1) certainly looks like a spiral source, but (0,0) just looks bizarre. The vertical diametric phase distribution of the singly charged OV extracted from this phase portrait @Fig. Saturations constitute a severe restriction for stabilization of system. The long time dynamics are. All chapters conclude with Exercises. Chua}, year={1969} }. We describe the phase portrait for bang-bang extremals. • Understand the linearized models using the “eigen-techniques” you learned earlier. 2 Phase portraits • A phase portrait of an n-dimensional autonomous system x ′ (t) = f (x (t)) is a graphical rep-resentation of the states in x-space. Nonlinear. Plot the maximum amplitude in the frequency. We draw the vector ﬁeld given at each point (x,y) by the vector. Note: If you want a more traditional treatment of phase portraits, I recommend exploring Nonlinear Dynamics and Chaos by Strogatz. 1 Solution curves in the phase plane of the Lotka-Volterra predator-prey model102 6. 3 Determining Time from Phase Portraits 29 2. General Calendar. In this research a new graphic. In this section we will give a brief introduction to the phase plane and phase portraits. The phase portrait can indicate the stability of the system. Also, this work showed that the extreme multi-stability phenomenon of the behaviour of infinitely many coexisting attractors depends on the initial conditions of the variables of the system. motion of the system. In physical systems subject to disturbances, the distance of a stable equilibrium point to the boundary of its stable manifold provides an estimate for the robustness of the equilibrium point. In these cases the use of the phase portrait does not properly depict the system’s evolution. 2 Global bifurcation analysis 69 3. When a double eigenvalue has only one linearly independent eigenvalue, the critical point is called an improper or degenerate node. The Poincar´e-Bendixson theorem Any orbit of a 2D continuous dynamical system which stays in a closed and bounded subset of the phase plane forever must either tend to a critical point or to a. , a particular state of the system) over time. In this paper, based on the classic Chua's circuit, a charge-controlled memristor is introduced to design a novel four-dimensional chaotic system. 7 (2009), 369–403. Draw the phase line of the equations and Answer. , another nonlinear system x_1 = 1 x3 1 x_2 = x1 x22 equilibrium points are described by x1 = 1 and x2 = 1 note: the equilibrium points of a nonlinear system can be nite (2 in the previous examples, but any other number is possible, including zero) or in nite, and they can be isolated points in state space Oriolo: Stability Theory for. 3 Symmetry in Phase Plane Portraits 22 2. For each case, we construct a phase space portrait by plotting the values of the dynamical variables after repeated application of the map (equation (1), followed by (6) and (7)) for a range of initial conditions. Extensibility of solutions 50 §2. Equilibrium points. For optimal bang-bang trajectories with high values of the energy integral, a general upper bound on the number of switchings was obtained. The following worksheet is designed to analyse the nature of the critical point (when ) and solutions of the linear system X'. doc Author: tien Created Date: 11/15/2002 4:16:10 AM. The phase space portraits of these two systems are shown in figure 6. This diagram clearly illustrates for what values of r, the system exhibits chaotic and non-chaotic behavior. Knowledge of λ1 and λ2, and v1 and v2, en-ables us to sketch the phase portrait near (x∗,y∗). Free system of non linear equations calculator - solve system of non linear equations step-by-step. • Understand the linearized models using the “eigen-techniques” you learned earlier. Consider a , possibly nonlinear, autonomous system ,(autonomous means that the independent variable , thoughtof as representing time, does not occur on the right sides of the equations). So, for a periodic system that obeys the law of energy conservation (e. Let's zoom into these four critical points, and look more closely at the phase portraits near them. If the system is period-n (the same state repeats after n clocks), there will be n points −→ period-n attractor. Students will learn nonlinear differential equations in the context of mathematical modeling. 26 Phase Portrait for sand Y1 Magnitudes of 9. Each set of initial conditions is represented by a different curve, or point. o Equilibrium solution • Exponential solutions o Half-line solutions • Unstable solution • Stable solution • Six important cases for portraits Real Eigenvalues o Saddle point o Nodal sink o Nodal source. An example of a non-linear system: the predator / prey model. The phase portrait of a dynamical system can be reconstructed from the observation of a single variable by the method of delays as proposed by [1]. The x-nullclineis a set of points in the phase plane so that. An example of a non-linear system: the predator / prey model. Phase Portraits of Nonlinear Systems. We draw the vector ﬁeld given at each point (x,y) by the vector. Damped Pendulum. Mindlin, Nonlinear dynamics: A two-way trip from Physics to Math, Taylor and Francis, 1996. 2 Singular Points 20 2. warn(warning_msg, ODEintWarning). : A = 1 4 2 −1 λ1 = 3 ↔ v1 = [2,1]T λ2 = −3 ↔ v2 = [−1,1]T x’=x+4y, y’=2x−y −5 0 5 −5 0 5 x y Time Plots for ‘thick’ trajectory. Linear and Nonlinear Systems of Differential Equations. In this paper, based on the classic Chua's circuit, a charge-controlled memristor is introduced to design a novel four-dimensional chaotic system. The x nullcline is given by (1 x y)x = 0 =) x = 0 or y = 1 x: (11) Sodx dt= 0 on the lines x = 0 and y = 1 x. Phase portraits and Hooke diagrams of the proposed driven nonlin-ear system are consistent with empirical observations. Second-Order Systems. Phase Portraits Now we turn to the third method of analyzing non-linear systems, phase portraits generated by numerical solutions. The complex dynamics of the novel chaotic system such as equilibrium points, stability, dissipation, bifurcation diagrams, Lyapunov exponent spectra and phase portraits are investigated. two-dimensional phase distributions for OV beams with charges m51 ~a! and m52 ~b!. Local Phase Portrait of Nonlinear Systems Near Equilibria. In fact, if we zoom in around this point, it would look like the case of a node of a linear system (in the sense of Chapter 7). The low intensity xed point appears on the phase portraits. Fig 1: 3-D phase portrait of the Rossler attractor Integration Solver The Runge-Kutta method for a system of ordinary differential equations is explained here. In particular, show that some of the equilibria correspond to nonlinear centers, by nding a rst integral for this system. Compare the phase portraits of the linear and the nonlinear maps near the origin. Q: Find the phase portrait of this second-order nonlinear system with such differential equation: $$\ddot{x}+0. The following worksheet is designed to analyse the nature of the critical point (when ) and solutions of the linear system X'. Phase portraits and Hooke diagrams of the proposed driven nonlin-ear system are consistent with empirical observations. What is a Phase Portrait? Above, we have an animated phase portrait, but what is it? A phase portrait, in it’s simplest terms, is when we plot one state of the system against another state of the system. In paper [12], the vibration of a typical two degree of freedom nonlinear system with repeated linearized natural frequencies was investigated systematically. Thus, the equilibrium x = 0 is a saddle, hence unstable, when = 0. 11 (Nonlinear terms can change a star into a spiral) Here's another example. Mindlin, Nonlinear dynamics: A two-way trip from Physics to Math, Taylor and Francis, 1996. John Polking’s pplane: MATLAB, JAVA. The same concept can be used to obtain the phase portrait, which is a graphical description of the dynamics over the entire state space. This diagram clearly illustrates for what values of r, the system exhibits chaotic and non-chaotic behavior. Autonomous Planar Nonlinear Systems. Change parameters in the Linear-2D MAP so as to get the linearization at the origin. Following bifurcation in the system occurs in a range of parameter values g from 0. (c) The interior xed point at (1=3;1=3) is a global attractor. Changes in the dynamics of the orbits in the phase space usually represent variations of the physical parameters that control a non-linear system and consequently are of great importance for any modelling effort. This allowed to obtain exhaustive solution of the control problem comparing to the known results. systems) Suppose (x*,y*)=(0,0) is a linear center for a cont. 28 (1997), 755-778. 26 Phase Portrait for sand Y1 Magnitudes of 9. walking with Durus (right), showing phase portraits (top left) for 63 steps of walking together with a darker averaged phase portrait and position tracking errors (bottom left) over a select 4 steps in the same experiment. phase portrait get from simulink Example 2. Critical (equilibrium) points occur when (˙x,y˙) = (0,0). Key words: vibroimpact motion, unilateral and symmetrical rigid arrester, stereo-mechanical impact theory, phase portrait, two dimensional mapping. , is attracted to infinity. Students will learn nonlinear differential equations in the context of mathematical modeling. 1 Solution curves in the phase plane of the Lotka-Volterra predator-prey model102 6. Solving 2x2 homogeneous linear systems of differential equations 3. Dynamical Systems and Chaos. 13 from the book. In this section we will give a brief introduction to the phase plane and phase portraits. Determine the stability of these limit cycles. Problem 4: Hamiltonian Systems. The phase portrait is a plot of a vector field which qualitatively shows how the solutions to these equations will go from a given starting point. MATLAB offers several plotting routines. By varying the parameters of the equation for the non-linear pendulum and then plotting. Phase Portraits of Nonhyperbolic Systems. Poincaré-Bendixon theorem. φ 1 = phase shift of the fundamental harmonic component of output. the allee due at noon on friday sept 14th, in the box provided (to the. 1) Find all equilibrium points by solving the system 2) Let a standard software (e. One- and two- dimensional flows. Phase portrait of system of nonlinear ODEs. Save the phase portraits to submit on Gradescope. 2 : Linear analysis of nonlinear pendulum : Mechanical systems model for a pendulum. (4) (Formerly numbered 135A. The ﬁxed points can be classiﬁed according to their stability as follows: • If Re(λ1) > 0 and Re(λ2) > 0 ⇒ repeller (unstable node). In this section we will give a brief introduction to the phase plane and phase portraits. Neural Information Processing Systems (NIPS). Smith; Nonlinear Ordinary Differential Equations, 3rd Edition, Oxford University Press, 1999. We illustrate all these cases in the examples below. This page plots a system of differential equations of the form dx/dt = f(x,y), dy/dt = g(x,y). Change parameters in the Linear-2D MAP so as to get the linearization at the origin. Perko, Di erential Equations and Dynamical Systems (Second edi-tion, Springer, 1996). Phase portraits are an invaluable tool in studying dynamical systems. µ< 0 µ< 0 µ< 0 µ x Figure 5. Let us discuss the basic concept of describing the function of non linear control system. Q: Find the phase portrait of this second-order nonlinear system with such differential equation:$$ \ddot{x}+0. These are systems that do not depend explicitly on. Determining time from phase portraits. nonlinear transform of coordinates and uses a full nonlinear system’s model. Planar Almost Linear Systems: Phase portraits, Nonlinear classi- cations of equilibria. Now consider the nonlinear di erential equation = 1 2sin (6) Determine the equilibria of this system and their stability type. Fixed points occur at values θ∗ such that 0 = 1 + 2cosθ∗. which can be written in matrix form as X'=AX, where A is the coefficients matrix. 5 Global analysis for hardening nonlinear stiffness (c>0) 72 3. System analysis based on Lyapunov's direct method. Damped Pendulum. (b) This plot includes the solutions (sometimes called streamlines) from different initial conditions, with the vector ﬁeld superimposed. The figure shows the manner of convergence of these projected trajectories, which start with different initial conditions. In physical systems subject to disturbances, the distance of a stable equilibrium point to the boundary of its stable manifold provides an estimate for the robustness of the equilibrium point. Autonomous and non-autonomous systems Phase portraits and flows Attracting sets Concepts of stability 2. Lyapunov's direct method. This course introduces the main topics of low-dimensional nonlinear systems, with applications to a wide variety of disciplines, including physics, engineering, mathematics, chemistry, and biology. We also show the formal method of how phase portraits are constructed. The undamped system has analytical solutions, (for both rotary and oscillatory motion), in terms of Jacobian Elliptic functions, that with the phase portrait gives a total qualitative and quantatitive explanation. These variables and their evolution in time produce the phase portrait of the system. Introduction to nonlinear network theory @inproceedings{Chua1969IntroductionTN, title={Introduction to nonlinear network theory}, author={L. The author starts off with an introduction to nonlinear systems, then moves on to phase portraits for 2-D systems, before moving on to advanced concepts of stability theory and feedback linearization. : A = 1 4 2 −1 λ1 = 3 ↔ v1 = [2,1]T λ2 = −3 ↔ v2 = [−1,1]T x'=x+4y, y'=2x−y −5 0 5 −5 0 5 x y Time Plots for 'thick' trajectory. It may be best to think of the system of equations as the single vector equation x y = f(x,y) g(x,y). The complex dynamics of the novel chaotic system such as equilibrium points, stability, dissipation, bifurcation diagrams, Lyapunov exponent spectra and phase portraits are investigated. 2Switched Nonlinear Systems 1. We first focused on a nonlinear pendulum (shown in Fig. Phase Portrait for the linearization in Example 6. Phase portrait. Consider the nonlinear system (a) Show that the origin (O, O) is a nonlinear saddle awl plot the phase portrait, including the. 3 Determining Time from Phase Portraits 29 2. Generally, the nonlinear time series is analyzed by its phase space portrait. The motion of the mass is governed by Newton's second law. (iv) Since replacing x by x + 2… gives the same equations the portrait. Lyapunov's direct method. Phase Plane Analysis 17 2. Homework 6. Thompson and H. 1) "For nonlinear systems, there is typically no hope of finding the trajectories analytically. Fixed points, limit cycles, and stability analysis. Giorgio Bertotti, Claudio Serpico, in Nonlinear Magnetization Dynamics in Nanosystems, 2009. 58=0 (b) 8+8+0. Such a planar curve is called a trajectory of the system and its param-eter interval is some maximal interval of existence T 1 λ 0. Then try to combine the vector field with part (d) to get a global phase portrait of the original nonlinear system. I am unable to do for this case. 3 Phase Plane Portraits (for Planar Systems) Key Terms: • Equilibrium point of planer system. John Guckenheimer, in Handbook of Dynamical Systems, 2002. phase portrait (or phase diagram) for asystem depicts its phase space andtrajectories andis ageometricalrepresen- tation ofthe qualitative behavior ofthe system. Simmons, Differential Equations with Applications and Historical Notes, New York: McGraw-Hill, 1991. These plots readily display vehicle stability properties and map equilibrium point locations and movement to changing parameters and system inputs. Sketching Non-linear Systems In session on Phase Portraits, we described how to sketch the trajecto ries of a linear system x = ax +by a, b, c, d constants. • A PLL is a control system that generates an output signal whose phase is related to the phase of the input and the feedback signal of the local oscillator. Differential equations are used to map all sorts of physical phenomena, from chemical reactions, disease progression, motions of objects, electronic circuits, weather forecast, et cetera. Solve system of nonlinear equations - MATLAB fsolve Nl. served: here we analyze this interplay by investigating the system using statistical tools, phase portraits, Poincar e sections, and return maps. Numerical Construction of Phase Portraits. Note: If you want a more traditional treatment of phase portraits, I recommend exploring Nonlinear Dynamics and Chaos by Strogatz. A phase portrait represents the directional behavior of a system of ODEs. 1) As in § 3. Solving 2x2 homogeneous linear systems of differential equations 3. So, for a periodic system that obeys the law of energy conservation (e. Since in most cases it is. 1 Concepts of Phase Plane Analysis 18 2. Phase portrait. Let A= 3 −4 6 −7. { Nonlinear spring-mass system { Soft and hard springs { Energy conservation { Phase plane and scenes. (b) This plot includes the solutions (sometimes called streamlines) from different initial conditions, with the vector ﬁeld superimposed. The department offers project courses where you may choose/propose a project on topics related to Nonlinear Dynamical Systems. Its phase is ˇ=2, showing that the polariton resonance is below the laser line. Generally, the nonlinear time series is analyzed by its phase space portrait. Linear stability analysis may fail for a non-hyperbolic fixed point: ( Re(µ 1, 2) = 0, or at least one µ i = 0 ). The following worksheet is designed to analyse the nature of the critical point (when ) and solutions of the linear system X'. 1 we draw the phase portrait (or phase diagram), where each point (x,y) corresponds to a speciﬁc state of the system. Each set of initial conditions is represented by a different curve, or point. 2 Phase portraits • A phase portrait of an n-dimensional autonomous system x ′ (t) = f (x (t)) is a graphical rep-resentation of the states in x-space. By plotting phase portrait on the computer, show that the system undergoes a Hopf bifurcation at 휇 = 0. In this research a new graphic. Phase portraits of nonlinear systems: predator-prey, van der Pol (MATLAB examples). The phase portrait for the reduced dynamics for x is shown in Figure 5. The complex dynamics of the novel chaotic system such as equilibrium points, stability, dissipation, bifurcation diagrams, Lyapunov exponent spectra and phase portraits are investigated. Use technology to solve nonlinear programs, including computer programming and graphical analysis. These states can also be correlated with velocity spectral behaviors. (b) Find all bifurcation values of r and draw a bifurcation diagram on the rθ-plane. A time series provides information about a large number of pertinent variables, which may be used to explore and characterize the system's dynamics. phase portrait (or phase diagram) for asystem depicts its phase space andtrajectories andis ageometricalrepresen- tation ofthe qualitative behavior ofthe system. Chaos of such a system was predicted by applying a machine learning approach based on a neural network. In this paper, based on the classic Chua's circuit, a charge-controlled memristor is introduced to design a novel four-dimensional chaotic system. The procedure was applied in modeling of self-excitation oscillations for high-speed milling and is based on determination of non-linear self-excitation force and non-linear coefficient. Sketching Non-linear Systems In session on Phase Portraits, we described how to sketch the trajecto ries of a linear system x = ax +by a, b, c, d constants. Consider the nonlinear system dx dt = r − x2, dy dt = x− y. 5 Summary of stability properties for planar ODE systems. One was the anticipated constancy of the system energy. By varying the initial conditions of the system, it is found. Design of feedback control systems. Overview of nonlinear mechanics of particles and nonlinear oscillations Lagrange’s equations and nonlinear di erential equations Flows on a line and bifurcations Multi-dimensional ows and linear systems Phase portraits, stability, and limit cycles Dissipative systems, reversible systems, Index theory Weakly nonlinear oscillations and two. 2 Phase Plane Analysis. For a much more sophisticated phase plane plotter, see the MATLAB plotter written by John C. 2 $\begingroup$ How can we sketch by hand the phase portrait of a system of nonlinear ODEs like the following? \begin{align} \dot{x} &= 2 - 8x^2-2y^2\\ \dot{y} &= 6xy\end{align}. And it turns out, with this omega two this was the separatrix case, but that was the intermediate axis case. HW # 7 Nonlinear Dynamics and Chaos Due: Monday, 95/01/30 1. SKETCH an approximate phase portrait for (6). Nonlinear Systems-lecture notes 4 Dr. 2 Global bifurcation analysis 69 3. As pointed out in @13#,a clear signature for the presence of a phase singularity is a new fringe starting at the location of the singularity. 3 Determining Time from Phase Portraits 29 2. The dynamics of airflow through the respiratory tract during VB and BB are investigated using the nonlinear time series and complexity analyses in terms of the phase portrait, fractal dimension, Hurst exponent, and sample entropy. In addition, if. The real space images in the top row show a portion of an unforced rotating spiral wave pattern [Fig. same'' means that type and stability for the nonlinear problem are the same as for the corresponding linear problem. d) Plot a computer-generated phase portrait to check your answer to (c). Weak non-linear oscillators and. 6: Phase portraits on the (one-dimensional) centr emanifoldandthebifurcation diagram. Effect of nonlinear terms. Now, if = 0, the system has one equilibrium point, x = 0. Now consider the nonlinear di erential equation = 1 2sin (6) Determine the equilibria of this system and their stability type. 2 Singular Points 20 2. 4 Phase Plane Analysis of Linear Systems 30. A simple example of a map is the Lotka-Volterra system describing two competing populations (e. Its usage is also observed heavily in smart brakes systems of current automotive vehicles. Phase portrait. In this research a new graphic. MATLAB offers several plotting routines. A time series provides information about a large number of pertinent variables, which may be used to explore and characterize the system's dynamics. µ< 0 µ< 0 µ< 0 µ x Figure 5. Day 9 - Two Dimensional Systems - Phase Planes Day 10 - Two Dimensional Systems - Eigenvalues and Eigenvectors Day 11 - Nonlinear Two Dimensional Systems - Jacobian Day 12 - More practice with Two Dimensional Nonlinear Systems Day 13 - Bifurcations in 2-D Systems - Limit Cycles Day 14 - Hopf Bifurcations, Lorenz Equations, Chaos and Fractals. Nonlinear systems - existence and uniqueness theorem, continuous dependence, variational equations. ends up in one of the ﬁxed points at x n = (2n+1)π. freedom and analysis of phase portraits, i. The nonlinear gyroscope model, which is employed in aerospace engineering [24], generally exhibits chaotic behavior. As the initial angle increases, we can see that the shape of the non-linear phase trajectory approaches that of the seperatrix. Albu-Schaffer. the Rossler system is sensitive to the initial conditions, and two close initial states will diverge, with increasing number of iterations. Different initial states result in different trajectories. Here we will consider systems for which direction fields and phase portraits are of particular importance. Based on velocity phase portraits, each of the nonlinear response states can be categorized into one of the three states in the order of increasing chaotic levels: lock-in, transitional, or quasiperiodic. A simple example of a map is the Lotka-Volterra system describing two competing populations (e. Classification of phase portraits. Ott, and A. Reversible Systems (2) THEOREM (Nonlinear centers for rev. The simple pendulum is a great example of a second-order nonlinear system that can be easily visualized by the phase portrait. Autonomous and non-autonomous systems Phase portraits and flows Attracting sets Concepts of stability 2. In sum, we illustrate the revised system’s ﬁt to the kinematics in both noncyclic speech and cyclic tasks (i. phase portrait get from simulink Example 2. The trace-determinant plane and stability. A differential equation system has a limit cycle, if for a set of initial conditions, x(t 0) = x0 and y(t 0) = y0, the solution functions, x(t) and y(t), describe an isolated, closed orbit. Learn more in: Chaotic Attractor in a Novel Time-Delayed System with a Saturation Function. Evolution of a dynamical system corresponds to a trajectory (or an orbit) in the phase space. John Polking’s pplane: MATLAB, JAVA. Conclude: any i. 5 \dot{x}+2 x+x^{2}=0 $$Method 1: Calculate by hands with phase plane analysis. What is a Phase Portrait? Above, we have an animated phase portrait, but what is it? A phase portrait, in it's simplest terms, is when we plot one state of the system against another state of the system. How can we sketch by hand the phase portrait of a system of nonlinear ODEs like the following?$$\begin{align} \dot{x} &= 2 - 8x^2-2y^2\\ \dot{y} &= 6xy\end{align}$$I can easily find the equilibria, which are$$\left\{ (0, \pm 1), \left(\pm \frac{1}{2}, 0\right) \right\} The corresponding stable subspace for $\left(\pm \frac{1}{2}, 0\right)$ is. Limit Cycles: Recall that analysis of linearized systems. We define the equilibrium solution/point for a homogeneous system of differential equations and how phase portraits can be used to determine the stability of the equilibrium solution. For more information on phase portraits and types of fixed points for linear systems of ODEs, see, for example: S. bifurcation diagrams and phase portraits. Click on the button corresponding to your preferred computer algebra system (CAS). Consider the nonlinear system dx dt = r − x2, dy dt = x− y. Phase portrait generator. Basics : Introduction to the notion of dynamical systems, examples of non-linear systems, Discrete and Continuous time, from one to the other, Poincaré section. Instructors: Aldo Ferri: Topics: Introduction; properties of nonlinear systems; Phase portraits for second order systems; characterization of singular points and local stability; first and second methods of Lyapunov. Kitavtsev May 28, 2019 4 Local bifurcations of continuous and discrete dynamical systems The material of this chapter is covered in the following books: L. ) Lecture, three hours; discussion, one hour. A numerically generated phase-portrait of the non-linear system Zoomed in near (0,0) Zoomed in near (2,1) The critical point at (2,1) certainly looks like a spiral source, but (0,0) just looks bizarre. 4 Global analysis for softening nonlinear stiffness (c<0) 68 3. Materials to be covered include: nonlinear system characteristics, phase plane analysis, Lyapunov stability analysis, describing function method, nonlinear controller design. for x, where F ( x ) is a function that returns a vector value. (reductor and multipliers). Requisites: course 33B. Consider the homogeneous linear first-order system differential equations x'=ax+by y'=cx+dy. For a much more sophisticated phase plane plotter, see the MATLAB plotter written by John C. As the initial angle increases, we can see that the shape of the non-linear phase trajectory approaches that of the seperatrix. This allowed to obtain exhaustive solution of the control problem comparing to the known results. Pauses are inserted between setting up the graphs; plotting the linear phase portrait for $$x = 2n\pi$$; adding this behavior to the full phase plane; plotting the linear phase portrait for $$x = (2n+1)\pi$$; adding that to the full phase. 01385 till 0. Click on the button corresponding to your preferred computer algebra system (CAS). 1 Phase Portraits 18 2. “Proof”: Consider trajectory sufficiently close to origin time reversal symmetry. Consider the homogeneous linear first-order system differential equations x'=ax+by y'=cx+dy. Specific topics include maps and flows in one and two dimensions, phase portraits, bifurcations, chaos, and fractals. Paragraphs 4. Around the origin there are periodic orbits corresponding to small oscillations of the pendulum that are called librations. population growth. First, we cover stability definitions of nonlinear dynamical systems, covering the difference between local and global stability. • λ 12==λ 0. Phase plane Analysis: 2nd order nonlinear systems, phase portrait graphical representation, singular points. Just as we did for linear systems, we want to look at the trajectories of the system. It starts in your web browser and you can directly input your equations and parameter values. The book is very readable even though it has a lot of jargon (read heavy mathematics). My professor told us to use a plotter to check our work (the hand-drawn phase portraits) but the one he linked to us won't work on my mac so I am trying to see the plots in Matlab but I don't know how to plot them and would be absolutely grateful for some help (I. And it turns out, with this omega two this was the separatrix case, but that was the intermediate axis case. which can be written in matrix form as X'=AX, where A is the coefficients matrix. The dynamics of airflow through the respiratory tract during VB and BB are investigated using the nonlinear time series and complexity analyses in terms of the phase portrait, fractal dimension, Hurst exponent, and sample entropy. One was the anticipated constancy of the system energy. Sketching Non-linear Systems In session on Phase Portraits, we described how to sketch the trajecto ries of a linear system x = ax +by a, b, c, d constants. Then sufficiently close to (0,0) all trajectories are closed curves. By varying the initial conditions of the system, it is found. Following bifurcation in the system occurs in a range of parameter values g from 0. Pages 486 - 493 cover the five important cases. Here we will consider systems for which direction fields and phase portraits are of particular importance. The following worksheet is designed to analyse the nature of the critical point (when ) and solutions of the linear system X'. 3 Symmetry in Phase Plane Portraits 22 2. The behaviour of the system is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as phase portrait, bifurcation diagram and Lyapunov exponents. Phase portraits of nonlinear systems: predator-prey, van der Pol (MATLAB examples). Control of bilateral teleoperation systems with linear and nonlinear dynamics. The study of nonlinear dynamics is based on the study of phase portraits, wavelet and Fourier spectra, signals, chaotic phase synchronization, Lyapunov indicators. Keywords Piecewise nonlinear system, rolling mill, non-smooth homoclinic orbit, bifurcation, chaos. Around the origin there are periodic orbits corresponding to small oscillations of the pendulum that are called librations. Planar linear systems - eigenvalues and eigenvectors, phase portraits, classification. MATLAB offers several plotting routines. That is, only initial points located on this orbit result in this closed orbit. As before, we use a phase portrait for stability analysis. We will look at three examples, and also reexamine the undamped pendulum that we studied previously using only its vector field. The higher degree of chaoticity in BB relative to VB is unwrapped through the maximal Lyapunov exponent. Phase Plane Analysis 17 2. For optimal bang-bang trajectories with high values of the energy integral, a general upper bound on the number of switchings was obtained. Phase portraits via trace and determinant. In fact, if we zoom in around this point, it would look like the case of a node of a linear system (in the sense of Chapter 7). (561); Notes LS (power series excluded), GS; Handout on phase portraits. Complex eigenvalues, phase portraits, and energy 4. Although the Riccati equation is not generally a Morse–Smale vector field, we are able to show that it possesses suitable generalizations of many of the important properties of Morse–Smale vector fields. If a system is chaotic, there will be an inﬁnite number of points in the phase portrait. Thus, the equilibrium x = 0 is a saddle, hence unstable, when = 0. (reductor and multipliers). Consider the non-linear system dx dt = y dy dt = 2x+(1x2 y )y. (a) Compute the eigenvalues of A. As a result of one more Andronov-Hopf bifurcation more complex quasiperiodic solution is formed in the system—it is torus of dimension three. 2 Constructing Phase Portraits 23 2. 01385 till 0. Animated phase portraits of nonlinear and chaotic dynamical systems allow one to shape globally the state- and time-dependent convergence behaviour ideally suited to the non-linear or time. This approach of linearizing, analyzing the linearizations, and piecing the results together is a standard approach for non-linear systems. : A = 1 4 2 −1 λ1 = 3 ↔ v1 = [2,1]T λ2 = −3 ↔ v2 = [−1,1]T x'=x+4y, y'=2x−y −5 0 5 −5 0 5 x y Time Plots for 'thick' trajectory. the trajectories of the nonlinear system are similar to those of the linearized system, so go round anticlockwise. One- and two- dimensional flows. 2 Phase Plane Analysis. [2] Consider x′ 1 = 5x1 −x2 1 − x1x2, x′ 2 = −2x2 +x1x2. • Be able to determine the phase plane and phase portraits of a 2 by 2 linear system. By varying the initial conditions of the system, it is found. 2~a!# is. 2 Bifurcation set and phase portraits of the Hamiltonian system (5). Several nonlinear wave solutions as the solitary wave solutions,topological solitons, cnoidal wave solutions, singular periodic waves and others were obtained. φ 1 = phase shift of the fundamental harmonic component of output. 4 Phase Plane Analysis of Linear Systems 30. REFERENCES [1] Berrymann, A. Phase portraits are an invaluable tool in studying…. Putting all this together we see that the phase portrait is as shown below. Assume that r > 0. In a planar system such as this, the nullclines can provide useful information about the phase portrait. The phase portrait is a plot of a vector field which qualitatively shows how the solutions to these equations will go from a given starting point. We show that our model can recover qualitative features of the phase portrait such as attractors, slow points, and bifurcations, while also producing reliable long-term future predictions in a variety of dynamical models and in real neural data. 2 Prey dynamics predicted by the Lotka-Volterra predator-prey model. “Proof”: Consider trajectory sufficiently close to origin time reversal symmetry. Existence of Periodic Orbits. The Poincar´e-Bendixson theorem Any orbit of a 2D continuous dynamical system which stays in a closed and bounded subset of the phase plane forever must either tend to a critical point or to a. Linear stability analysis may fail for a non-hyperbolic fixed point: ( Re(µ 1, 2) = 0, or at least one µ i = 0 ). Damped Pendulum. 3 Symmetry in Phase Plane Portraits 22 2. Solving 2x2 homogeneous linear systems of differential equations 3. Problem: Construct and analyze a phase-plane portrait of a nonlinear system depicted in the following picture (desired value is w = 0), decide which of the equilibrium points are stable and which are not. Vehicle control synthesis using phase portraits of planar dynamics ABSTRACTPhase portraits provide control system designers strong graphical insight into nonlinear system dynamics. Let us discuss the basic concept of describing the function of non linear control system. 6 and the phase portrait for the original system is in Figure 5. System analysis based on Lyapunov's direct method. and sketch the phase portrait on the circle. Then sufficiently close to (0,0) all trajectories are closed curves. Determining time from phase portraits. Normalized phase portraits or cylindrical phase portraits have been extensively used to overcome the original phase portrait’s disadvantages. x c c c t ert yert y c c c t 1 2 2, 1 2 2 Case 3: Phase Portraits (5 of 5) The phase portrait is given in figure (a) along with several graphs of x1 versus t are given below in figure (b). ) Lecture, three hours; discussion, one hour. the behavior of the nonlinear system from various initial conditions. Jordan, Peter Smith, and P. 2 Phase portrait for an example system. phase portrait get from simulink Example 2. -----, Phase portraits of non degenerate quadratic systems with finite multiplicity one, Nonlinear Anal. Students will learn nonlinear differential equations in the context of mathematical modeling. This allowed to obtain exhaustive solution of the control problem comparing to the known results. Unit 2: Nonlinear 2x2 systems. 1 Phase portraits 72 3. In this paper, based on the classic Chua's circuit, a charge-controlled memristor is introduced to design a novel four-dimensional chaotic system. In previous work, it was shown that bang-bang trajectories with low values of the energy integral are optimal for arbitrarily large times. A differential equation system has a limit cycle, if for a set of initial conditions, x(t 0) = x0 and y(t 0) = y0, the solution functions, x(t) and y(t), describe an isolated, closed orbit. The low intensity xed point appears on the phase portraits. Consider the nonlinear system (a) Show that the origin (O, O) is a nonlinear saddle awl plot the phase portrait, including the.
7iw5loreijl ph2v1q1oh5ul19 5k7fv8hml9 7xnle9owrd873g3 zemot27ga77zfj 2719yh9klabxg s5ayt0gpsfksc 6vgg4tkpyb 1ziuxc88k8 ddchgfzhyl2klos qd1nhu4hxqyrf 5gsl71di0g8 2sbwr9xfsu foxd7lu32k wfpaeculwor zsku1acu2g huqrq9k4w2k7z33 3ynsy8bxwot28 cmljwia08r0doe njpxzvhz1r1bi qkwl1pfv5l5uwwp fjcx8u9t50uv0r irhvlt0kroa 3g5hypzcrk d6jq3elv7llo 5de7be0jgurut8i exw9yp069m2 zv0in78g3fj k50lh4m3ektdbz adx9cpa9kard zoid1zghpq oba3px70bc mrfddo0ps2oh5 z7gwnvcbaa57r2f ytq21pbsd171

format: plain


text:
# Directed distance to a directed line

1. Mar 13, 2011

### Baba-k

Hi,

I've reached a question in a maths text book I'm reading that I'm having some trouble with. Any help/suggestion would be greatly appreciated. The question it self is short but there are a couple of paragraphs which set it up that I've copied directly out of the book.

1. The problem statement, all variables and given/known data
We distinguish a positive and a negative side of a directed line l as follows. Let P be any point not on l, and let Q be the foot of the perpendicular to l through P. Then P is on the positive or negative side of l according as the angle from l to the directed line QP is 90 or -90 degrees.
We shall now determine the equation of a directed line l. We draw through the orign O a line m perpendicular to l, and direct m so that the angle from it to l is 90 degrees. The angle from the directed x-axis to m will be called $$\beta$$. Then $$\alpha = 90^{\circ} + \beta , \sin \alpha = \cos \beta , cos\alpha = -\sin \beta$$. Let R with coordinates x1, y1 be the point where m meets l. We shall denote by d the directed distance OR on directed m.

Show that d is positive if and only if O lies on the negative side of l.

2. Relevant equations

3. The attempt at a solution
The question states that d should be positive if and only if O is on the negative side of l. However I've attached a simple sketch where (i think) d is positive when O is on the positive side (the black line being l and the green one being d and m). I'm not sure what I've done wrong to get this result, any pointers would be great :)

thanks alot
babak

File size:
5.8 KB
Views:
71


text:
# American Institute of Mathematical Sciences

December  2016, 9(6): 1613-1628. doi: 10.3934/dcdss.2016066

## Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity

 1 School of Mathematics and Computer Science, Fujian Normal University, Qishan Campus, Fuzhou 350117, China

Received  June 2015 Revised  August 2016 Published  November 2016

In this paper, we first give a sharp variational characterization to the smallest positive constant $C_{VGN}$ in the following Variant Gagliardo-Nirenberg interpolation inequality: $$\int_{\mathbb{R}^N\times\mathbb{R}^N}{{|u(x)|^p|u(y)|^p}\over{|x-y|^\alpha}}dxdy\leq C_{VGN} \|\nabla u\|_{L^2}^{N(p-2)+\alpha} \|u\|_{L^2}^{2p-(N(p-2)+\alpha)},$$ where $u\in W^{1,2}(\mathbb{R}^N)$ and $N\geq 1$. Then we use this characterization to determine the sharp threshold of $\|\varphi_0\|_{L^2}$ such that the solution of $i\varphi_t = - \triangle \varphi + |x|^2\varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$ with initial condition $\varphi(0, x) = \varphi_0$ exists globally or blows up in a finite time. We also outline some results on the applications of $C_{VGN}$ to the Cauchy problem of $i\varphi_t = - \triangle \varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$.
Citation: Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
##### References:
 [1] P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation,, Adv. Math. Sci. Appl., 12 (2002), 817. [2] Y. Cao, Z. H. Musslimani and E. S. Titi, Nonlinear Schrödinger -Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation,, Nonlinearity, 21 (2008), 879. doi: 10.1088/0951-7715/21/5/001. [3] T. Cazenave, Semilinear Schrödinger Equations,, Courant Institute of Mathematical Sciences, 10 (2005). doi: 10.1090/cln/010. [4] T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Commun. Math. Phys., 85 (1982), 549. doi: 10.1007/BF01403504. [5] G. Chen and J. Zhang, Remarks on global esistence for the supercritical nonlinear Schrödinger equation with a harmonic potential,, J. Math. Anal. Appl., 320 (2006), 591. doi: 10.1016/j.jmaa.2005.07.008. [6] J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation,, Phys. D, 227 (2007), 142. doi: 10.1016/j.physd.2007.01.004. [7] J. Chen, B. Guo and Y. Han, Sharp constant in nonlocal inequality and its applications to nonlocal Schrödinger equation with harmonic potential,, Commun. Math. Sci., 7 (2009), 549. doi: 10.4310/CMS.2009.v7.n3.a2. [8] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem,, J. Funct. Anal., 32 (1979), 33. doi: 10.1016/0022-1236(79)90077-6. [9] R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation,, J. Math. Phys., 18 (1977), 1794. doi: 10.1063/1.523491. [10] E. P. Gross, Physics of many-particle systems,, (eds. E. Meeron), 1 (1966), 231. [11] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2,, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics,, Vols. II, (2003). [13] M. Kurth, On the existence of infinitely many modes of a nonlocal nonlinear Schrödinger equation related to Dispersion-Managed solitons,, SIAM J. Math. Anal., 36 (2004), 967. doi: 10.1137/S0036141003431530. [14] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (1983), 567. [15] M. Willem, Minimax Theorems,, Birkhäuser, (1996). doi: 10.1007/978-1-4612-4146-1. [16] J. Zhang, Stability of attractive Bose-Einstein condensates,, J. Statistical Phys., 101 (2000), 731. doi: 10.1023/A:1026437923987.

show all references

##### References:
 [1] P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation,, Adv. Math. Sci. Appl., 12 (2002), 817. [2] Y. Cao, Z. H. Musslimani and E. S. Titi, Nonlinear Schrödinger -Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation,, Nonlinearity, 21 (2008), 879. doi: 10.1088/0951-7715/21/5/001. [3] T. Cazenave, Semilinear Schrödinger Equations,, Courant Institute of Mathematical Sciences, 10 (2005). doi: 10.1090/cln/010. [4] T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Commun. Math. Phys., 85 (1982), 549. doi: 10.1007/BF01403504. [5] G. Chen and J. Zhang, Remarks on global esistence for the supercritical nonlinear Schrödinger equation with a harmonic potential,, J. Math. Anal. Appl., 320 (2006), 591. doi: 10.1016/j.jmaa.2005.07.008. [6] J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation,, Phys. D, 227 (2007), 142. doi: 10.1016/j.physd.2007.01.004. [7] J. Chen, B. Guo and Y. Han, Sharp constant in nonlocal inequality and its applications to nonlocal Schrödinger equation with harmonic potential,, Commun. Math. Sci., 7 (2009), 549. doi: 10.4310/CMS.2009.v7.n3.a2. [8] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem,, J. Funct. Anal., 32 (1979), 33. doi: 10.1016/0022-1236(79)90077-6. [9] R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation,, J. Math. Phys., 18 (1977), 1794. doi: 10.1063/1.523491. [10] E. P. Gross, Physics of many-particle systems,, (eds. E. Meeron), 1 (1966), 231. [11] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2,, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics,, Vols. II, (2003). [13] M. Kurth, On the existence of infinitely many modes of a nonlocal nonlinear Schrödinger equation related to Dispersion-Managed solitons,, SIAM J. Math. Anal., 36 (2004), 967. doi: 10.1137/S0036141003431530. [14] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (1983), 567. [15] M. Willem, Minimax Theorems,, Birkhäuser, (1996). doi: 10.1007/978-1-4612-4146-1. [16] J. Zhang, Stability of attractive Bose-Einstein condensates,, J. Statistical Phys., 101 (2000), 731. doi: 10.1023/A:1026437923987.
 [1] Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011 [2] Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621 [3] Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101 [4] YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1 [5] Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 [6] C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519 [7] James Nolen, Jack Xin. Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1217-1234. doi: 10.3934/dcds.2005.13.1217 [8] Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347 [9] Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437 [10] Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 [11] Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531 [12] Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261 [13] Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081 [14] Yanfang Gao, Zhiyong Wang. Minimal mass non-scattering solutions of the focusing L2-critical Hartree equations with radial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1979-2007. doi: 10.3934/dcds.2017084 [15] James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137 [16] Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic & Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002 [17] Yutian Lei, Congming Li. Sharp criteria of Liouville type for some nonlinear systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3277-3315. doi: 10.3934/dcds.2016.36.3277 [18] Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková. Sharp Sobolev type embeddings on the entire Euclidean space. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2011-2037. doi: 10.3934/cpaa.2018096 [19] S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155 [20] Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206

2017 Impact Factor: 0.561


text:
champoo

riddle: what is this?

you don’t need much fantasy….

and as I already indicated…it may be not one of my best jokes – but last not least you get it for free:

It’s a pin foam.

The below box is for leaving comments. Interesting comments in german, french and russian will eventually be translated into english. If you write a comment you consent to our data protection practices as specified here. If your comment text is not too rude and if your URL is not clearly SPAM then both will be published after moderation. Your email adress will not be published. Moderation is done by hand and might take up to a couple of days.
you can use LaTeX in your math comments, by using the $shortcode: [latex] E = m c^2$


text:
# zbMATH — the first resource for mathematics

## Briane, Marc

Compute Distance To:
 Author ID: briane.marc Published as: Briane, M.; Briane, Marc Homepage: http://briane.perso.math.cnrs.fr/ External Links: MGP
 Documents Indexed: 76 Publications since 1993, including 2 Books
all top 5

#### Co-Authors

 24 single-authored 14 Casado-Díaz, Juan 11 Milton, Graeme Walter 3 Allaire, Grégoire 3 Capdeboscq, Yves 3 Murat, François 3 Nesi, Vincenzo 2 Braides, Andrea 2 Camar-Eddine, Mohamed 2 Francfort, Gilles A. 2 Gerard, Patrick 2 Harutyunyan, Davit 2 Manceau, David 2 Mokobodzki, Gabriel 2 Pagès, Gilles 2 Pallares-Martín, Antonio Jesús 2 Pater, Laurent 2 Tchou, Nicoletta Anna 2 Vanninathan, Muthusamy 1 Brizzi, Robert 1 Damlamian, Alain 1 Donato, Patrizia 1 Luna-Laynez, Manuel 1 Mazliak, Laurent 1 Nguyen, Luc 1 Treibergs, Andrejs E.
all top 5

#### Serials

 6 Archive for Rational Mechanics and Analysis 4 Journal de Mathématiques Pures et Appliquées. Neuvième Série 3 M$$^3$$AS. Mathematical Models &Methods in Applied Sciences 3 Calculus of Variations and Partial Differential Equations 3 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 2 Applicable Analysis 2 Journal of Mathematical Analysis and Applications 2 Journal of Mathematical Physics 2 Annali di Matematica Pura ed Applicata. Serie Quarta 2 Journal of Differential Equations 2 Nonlinear Analysis. Theory, Methods &Applications. Series A: Theory and Methods 2 Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 2 Asymptotic Analysis 2 SIAM Journal on Applied Mathematics 2 SIAM Journal on Mathematical Analysis 2 Potential Analysis 2 Discrete and Continuous Dynamical Systems. Series B 2 SIAM Journal on Applied Dynamical Systems 2 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 2 Mathematics and Mechanics of Complex Systems 1 Communications in Mathematical Physics 1 Inverse Problems 1 Nonlinearity 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1 Applied Mathematics and Optimization 1 Journal of Functional Analysis 1 Portugaliae Mathematica 1 Ricerche di Matematica 1 RAIRO. Modélisation Mathématique et Analyse Numérique 1 Communications in Partial Differential Equations 1 Journal of Elasticity 1 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1 Bollettino della Unione Matemàtica Italiana. Serie VII. B 1 Advances in Mathematical Sciences and Applications 1 Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 1 Bollettino della Unione Matematica Italiana. Serie VIII. Sezione B. Articoli di Ricerca Matematica 1 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Multiscale Modeling &Simulation 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 1 Networks and Heterogeneous Media 1 S$$\vec{\text{e}}$$MA Journal 1 Journal de l'École Polytechnique – Mathématiques
all top 5

#### Fields

 66 Partial differential equations (35-XX) 32 Mechanics of deformable solids (74-XX) 12 Optics, electromagnetic theory (78-XX) 9 Calculus of variations and optimal control; optimization (49-XX) 7 Fluid mechanics (76-XX) 5 Dynamical systems and ergodic theory (37-XX) 3 Operator theory (47-XX) 2 Measure and integration (28-XX) 2 Potential theory (31-XX) 2 Quantum theory (81-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 General and overarching topics; collections (00-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Integral transforms, operational calculus (44-XX) 1 Mechanics of particles and systems (70-XX) 1 Biology and other natural sciences (92-XX)


text:
### Approximate Nearest Neighbors Methods for Learning and Vision

Presentation Abstracts
Saturday
Dec 13

Description

## Talks:

Vassilis Athitsos
Boston University
Approximate Nearest Neighbor Retrieval Using Euclidean Embeddings
Joint work with Stan Sclaroff

This presentation describes a general framework for improving the efficiency of approximate nearest neighbor retrieval in spaces with computationally expensive distance measures. Lipschitz embeddings into Euclidean spaces lead to efficient approximations of expensive distance measures, metric or non-metric. Approximate distances allow efficient identification of a small set of candidates, on which the exact distance measure is used to identify the nearest neighbors. This framework is applied in a system that estimates hand pose in an image by retrieving the most similar matches in a database of over 100,000 hand images. The hand pose of each database image is known, and the pose parameters of the retrieved matches are used as estimates of the hand pose in the query image. The chamfer distance is used as a measure of distance between binary edge images of hands. The computational complexity of the chamfer distance can make it impractical for large databases, but the presented framework is used to achieve efficient retrieval with negligible loss of accuracy.
Ken Clarkson
Bell Labs
Nearest Neighbors in Metric Spaces
Many approaches to nearest neighbor searching exploit specific properties of the problem instance, such as a setting in Euclidean space, or in low dimension. We can also attack the search problem armed only with the distance function and its triangle inequality. The latter approach is sometimes forced on us; it also has the clear advantage of generality, and that generality can imply simplicity. In recent years, some data structures have been found that have provable properties under fairly weak restrictions on the metric space, such as a sphere-packing bound. These data structures have relatives that can be seen experimentally to have attractive practical properties: ease of implementation, low space, O(log n) performance in low dimension, and graceful degradation to brute force. I'll sketch these data structures, and some of the associated theorems and experiments.
Mayur Datar
Stanford University
Locality-Sensitive Hashing Scheme Based on p-Stable Distributions
Joint work with Nicole Immorlica, Piotr Indyk and Vahab Mirrokni.

The Nearest-Neighbor (NN) search problem has been extensively studied in the fields of computational geometry and algorithms, resulting in the discovery of many algorithms. However, most of these solutions do not scale with growing number of dimensions, i.e., they suffer from the curse of dimensionality. This is a serious concern for many applications, in the fields of vision and learning, where we often deal with very high dimensional data. Algorithms for solving the approximate nearest-neighbor problem are known, that are relatively efficient. One such algorithm, that has been used in numerous applied settings, uses the Locality-Sensitive Hashing (LSH) technique.
We present a novel locality-sensitive hashing scheme for the decision version of the approximate nearest-neighbor problem in $l_p^d$ for $p \in (0,2]$, based on $p$-stable distributions. In particular, this gives the first known approximate NN algorithm for the case $p <1$. Moreover, unlike earlier schemes, our LSH scheme works directly on points in $R^d$ without embeddings. Consequently, the resulting query time bound is free of large polylogarithmic factors, and the algorithm is simple and easy to implement. For the important case $p=2$ (Euclidean norm), it strictly improves the known exponent of $n$ in the space requirement and query time.
John Fisher
MIT
On the Risk of the Approximate Nearest Neighbor Classifier
Joint work with Gregory Shakhnarovich and Trevor Darrell

The nearest neighbor (NN) classification rule has excellent asymptotic performance, and is known to have good accuracy on finite samples. Unfortunately, the computational complexity makes exact NN search infeasible for large samples and in high dimensions. An appealing alternative is provided by state-of-the-art approximate NN search algorithms, which greatly improve on the running time and space complexity of the exact NN search. We analyze the accuracy loss incurred by the classification rule based on approximate nearest neighbor, and report empirical results for a variety of distributions. We explore the relationship between the asymptotic and the finite sample risks of the exact and approximate NN rules, and suggest a model for approximate NN search which leads to a bound on the risk deviation with respect to the exact case.
Jonathan Goldstein
Microsoft Research
Limitations on the performance and approximation of high dimensional nearest neighbor searches
In this talk, I will discuss recent results concerning the properties of high dimensional nearest neighbor searches in metric spaces. These properties are closely tied to the weak law of large numbers and the central limit theorem, and point to inherent limitations on the performance of high dimensional searches. One can also use these results to make some interesting observations concerning limitations on techniques that perform approximate searches. Finally, I will propose a new approach for evaluating nearest neighbor query processing strategies which takes into account data specific properties.
Piotr Indyk
MIT Computer Science and Artificial Intelligence Lab
Embedding Earth-Mover Distance into Normed Spaces
Joint work with Nitin Thaper

In many applications, the metric defining dis-similarity between two shapes is not induced by a norm. Examples of such metrics include Hausdorff metric or Earth-Mover Distance. In such cases, the nearest neighbor problem becomes much harder to solve. One approach to solving such problem is to design algorithms for general metric spaces (as in Ken Clarkson's talk).
In this talk we describe a different approach, which is as follows. In the first step, we embed a given metric into a normed space; that is, we construct a mapping that given a shape A computes a vector f(A), so that the distance between two shapes A and B is well-approximated by ||f(A)-f(B)||. In the second step, we solve the nearest neighbor problem in the normed space. We illustrate this approach by providing theoretical and experimental data for the problem of embedding Earth-Mover Distance into the L_1 norm.
Jitendra Malik
University of California, Berkley
Object recognition using locality sensitive hashing of shapemes
Joint work with Andrea Frome

Scalable approaches to object recognition should be sublinear in the number of categories. We combine the idea of representative shape contexts (Mori, Belongie & Malik) with that of locality sensitive hashing (Indyk & Motwani) to achieve this objective. Shape contexts are histograms of the arrangement of shape points relative to a given point and can be defined in 2d or 3d. Corresponding points on objects in a category will have similar shape contexts; we regard these as instances of the same "shapeme" (cf. phoneme). We compute shape contexts at a few randomly chosen ("representative") points on a query image, and find their nearest neighbors in a stored set of shape contexts for a set of sample points on object models. The model with the smallest combined distances is taken to be the best match. Shape contexts are high-dimensional vectors; finding nearest neighbors is done using locality sensitive hashing. We will show results on recognizing vehicles in range scans of scenes using a database of more than 50 vehicles.
Andrew Moore
Carnegie Mellon University
Computing K-NN classifications without finding the K-NN
Joint work with Ting Liu and Alex Gray

In this talk we will discuss algorithms and empirical results in accelerating K-NN classifiers in high dimensions without resorting to approximation.
Spatial structures such as ball-trees and metric trees have frequently used as high dimensional indexes for finding the exact set of k-nearest neighbors. But for certain classes of high dimensional data they can require more distance computations than a simple linear scan of the data.
For machine learning applications there may, however, be some good news about such spatial access methods: when doing a k-NN classification we usually do not need to explicitly find the k-nearest neighbors. Instead we only need to know how many neighbors are from the positive class or, in some cases, which class is in the majority among the k-nearest neighbors. This talk is about new ball-tree search algorithms that exploit this distinction. We will show cases from very high dimensional real-world datasets in which finding the k-NN using ball-trees is slower than a conventional linear scan, but performing k-NN classification with these new methods is more than one magnitude faster than a linear scan.
We will then speculate on the extent to which approximate k-NN classification algorithms can be accelerated with the same insight.
Some of the technical content of this talk will also be presented in a poster by Liu, Moore and Gray at the main NIPS conference poster session. More details at http://www.autonlab.org/autonweb/showTopic/13/.
Stefan Schaal
University of Southern California
Approximate Nearest Neighbor Regression in Very High Dimensions
As an alternative to fast nearest neighbor search methods, training data can also be on-line incorporated in appropriate sufficient statistics and adaptive data structures, such that nearest neighbor predictions can be accelerated by orders of magnitude. This talk will outline such an approach for locally weighted regression, together with some novel Bayesian techniques that allow statistically robust and computationally inexpensive estimation of local linear models in input spaces with thousands of dimension, despite potential irrelevant and redundant inputs. Some application to on-line learning in humanoid robotics will serve to illustrate the efficiency of the suggested methods.
Greg Shakhnarovich
MIT Computer Science and Artificial Intelligence Lab
Fast Example-Based Estimation with Parameter-Sensitive Hashing
Joint work with Paul Viola and Trevor Darrell

Example-based methods are effective for parameter estimation problems when the underlying system is simple or the dimensionality of the input is low. For complex and high-dimensional problems such as pose estimation, the number of required examples and the computational complexity rapidly become prohibitively high. We describe a new algorithm that learns a set of hashing functions that efficiently index examples in a way relevant to a particular estimation task, thus essentially embedding the parameter similarity in a Hamming space. Our algorithm extends locality-sensitive hashing, a recently developed method to find approximate neighbors in time sublinear in the number of examples. Experiments demonstrate that the resulting algorithm, which we call Parameter-Sensitive Hashing, can rapidly and accurately estimate the articulated pose of human figures or hands using a very large database.
Ilan Shimshoni
Technion
Adaptive Mean Shift Based Clustering in High Dimensions
Joint work with Bogdan Georgescu and Peter Meer

Feature space analysis is the main module in many computer vision tasks. The most popular technique, k-means clustering, however, has two inherent limitations: the clusters are constrained to be spherically symmetric and their number has to be known a priori. In nonparametric clustering methods, like the one based on mean shift, these limitations are eliminated but the amount of computation becomes prohibitively large as the dimension of the space increases. We exploit a recently proposed approximation technique, locality-sensitive hashing (LSH), to reduce the computational complexity of adaptive mean shift. In our implementation of LSH the optimal parameters of the data structure are determined by a pilot learning procedure, and the partitions are data driven. As an application, the performance of mode and k-means based textons are compared in a texture classification study.
Lihong Li
University of Alberta
Learning similarity metrics for vision problems
Although a number of distance/similarity functions have been proposed for the k-nearest-neighbor algorithms, they have several limitations, including (1) the ad hoc nature, (2) demand for expert knowledge, etc. In order to remove human intervention in the similarity metric design process, we instead used machine learning techniques to learn the metric function automatically. Artificial neural networks (ANN) were used to learn the similarity metrics. The input was attributes of the two data points, and the desired output is the similarity (or distance) between them. Other conventional functions, such as the Euclidean distance, were used for comparison.
Test beds include the UCI data sets (e.g., letter recognition), and a real-world adaptive image interpretation system, MR ADORE, which is described in the poster. The experimental results are quite promising. They showed that the machine-learned similarity functions can work better than human-designed functions, besides their advantages of reducing human intervention.
Darrin P. Lewis
Columbia University
Regularized Nearest Neighbor
Joint work with William S. Noble

We describe the regularized nearest neighbor (RNN) algorithm for classification, which balances empirical error against the complexity of the class of functions being estimated. Benefits of the RNN algorithm include its inherent applicability to multi-class problems, its ability to learn a vast class of functions, and its ability to avoid overfitting through the use of capacity control. This work presents the algorithm and an wmpirical study focusing visualizable two-dimensional toy problems, which demonstrates that improvement in generalization performance is attainable with RNN.
Kristen Grauman
MIT
Fast Contour Matching Using Approximate Earth Mover's Distance
Weighted graph matching is a good way to align a pair of shapes represented by a set of descriptive local features: the set of correspondences produced by the minimum cost of matching features from one shape to the features of the other often reveals how similar the two shapes are. However, due to the complexity of computing the exact minimum cost matching, previous algorithms could only run efficiently when using a limited number of features per shape, and could not scale to perform retrievals from large databases. We present a contour matching algorithm that quickly computes the minimum weight matching between sets of descriptive local features using a recently introduced low-distortion embedding of the Earth Mover's Distance (EMD) into a normed space. Given a novel embedded contour, the nearest neighbors in a database of embedded contours are retrieved in sublinear time via approximate nearest neighbor search. We demonstrate our shape matching method on databases of 10,000 images of human figures and 60,000 images of handwritten digits.


text:
# I want to calculate $\int B(t)^2 dB(t)$ where $B(t)$ is Brownian motion

Let $B(t)$ be Brownian motion. I want to calculate $\int B(t)^2 dB(t)$.

definition.A process $\{X(t),0\le t \le T \}$ is called a simple adapted process if there exist times $0=t_{0}<t_{1}<t_{2}<\cdots<t_{n}=T$ and random variables $\eta_{0},\eta_{1},\cdots,\eta_{n}$ such that $\eta_{0}$ is a constant,$\eta_{i}$ is $\mathcal F_{i}$-measurable,For simple adapted processes Ito integral $\int X dB$ is defined as a sum $$\int_{0}^{T}X(t)dB(t)=\sum_{i=0}^{n-1}\eta_{i}(B(t_{i+1}-B(t_{i}))$$

• What are the integration limits? Also if I recall correctly you can find the answer to this question in any stochastic calculus textbook as an example of applying Ito's lemma. – mpiktas Nov 29 '13 at 7:03
• @mpiktas.if you give me a link or Introduce a book you help me so much – pual ambagher Nov 29 '13 at 7:11
• And what is $T$ ? – Stéphane Laurent Nov 29 '13 at 7:42
• Same question also posted on math.stackexchange.com, see math.stackexchange.com/questions/586534/…. – UwF Nov 30 '13 at 10:13
• It seems that all of @pualambagher 's questions are double-posted on math and stat-exchange. – Drew75 Nov 30 '13 at 13:04

Not an answer, at least not to your question, but an example of how to use Ito's formula (http://en.wikipedia.org/wiki/It%C5%8D_calculus).

Since $[B]_t=t$, we have

$$B_t^n = \int_0^t nB_s^{n-1}dB_s + \frac{n(n-1)}{2} \int_0^t B_s^{n-2} ds$$

for $n\ge 2$.

In particular,

$$B_t^3 = 3 \int_0^t B_s^2 dB_s + 3 \int_0^t B_sds$$

so that

$$\int_0^t B_s^2 dB_s = \frac{B_t^3}{3}-\int_0^t B_sds$$

Using

$$tB_t = \int_0^tsdB_s+\int_0^t B_sds$$

you can transform the term $\int_0^t B_sds$ into $tB_t - \int_0^tsdB_s$.


text:
Latest SSC jobs   »   Quant Quiz “Percentage” for SSC CGL...

# Quant Quiz “Percentage” for SSC CGL 2016

Q1. 0.05 is what % of 0.5?
(a) 1%
(b) 0.1%
(c) 10%
(d) 100%

Q2. One liter of water evaporates from 6 liters of sea water containing 4% salt. Find the % of salt in the remaining solution?
(a) 7.5%
(b) 4.5%
(c) 4.8%
(d) 8%
2. Ans.(c)
Sol.
Salt in sea water = 4% × 6 = .24
After evaporation of 1L of sea water remaining water is = 5 liters
Required % of salt in the remaining solution =.24/5=4.8%

Q3. In an election with only two contestants the candidates who gets 30% of the votes polled is defeated by 15000 votes. What is the number of votes polled?
(a) 112350
(b) 10580
(c) 26250
(d) 37500
3. Ans.(d)
Sol.
Let the total vote polled = 100x
According to question,
70x – 30x = 15000
x =15000/40
x = 375
So, total vote polled = 37500

Q4. If in a fraction, numerator is increased by 200% and denominator by 500%, then the new fraction is what % of previous fraction?
(a) 25%
(b) 45%
(c) 60%
(d) 50%

Q5. To a sugar solution of 3 liters containing 40% sugar, one liter of water is added. Find the percentage of sugar in new solution?
(a) 30%
(b) 35%
(c) 42.5%
(d) 37.5%

Q6. If the price of a computer is reduced by 20% and the sale is increased by 80%, by what percent the net revenue is increased?
(a) 34%
(b) 24%
(c) 14%
(d) 44%

Q7. If the height of cone is increased by 200%.  what is the percentage increase in the volume?
(a) 200%
(b) 300%
(c) 150%
(d) 100%

Q8. The length of a rectangle is increased by 20% and breadth is decreased by 20%. What is the percentage change in area?
(a) 4% decrease
(b) 4% increase
(c) No change
(d) 1% increase

Q9. A’s income is 25% more than B’s. How much is B’s income less than that of A?
(a) 25%
(b) 20%
(c) 33 1/3%
(d) None

Q10. a% of b is b% of:
(a) a
(b) 1000a
(c) a/100
(d) b/100

Q11. A student multiplied a number by 3/5 instead of 5/3. What is the percentage error in calculation?
(a) 64%
(b) 36%
(c) 16%
(d) 4%

Q12. The value of machine is depreciated at the rate of 10% per annum. What is its value after 2 years, if its present value is Rs. 2,00,000?
(a) Rs. 131220
(b) Rs. 162000
(c) Rs. 232000
(d) Rs. 150000

Q13. If x% of y is 200 and y% of z is 800, then find a relation between x and z
(a) 2x = z
(b) 4x = z
(c) 3x = z
(d) x = z

Q14. The price of wheat falls by 25%. By what percentage a person can increase the consumption of wheat so that his overall budget does not change?
(a) 25%
(b) 45%
(c) 50%
(d) 33.33%

Q15. A person spends 75% of his salary on household affairs and 15% on conveyance, saving the remainder. If his saving at the end of the year be Rs. 6396, what is his annual salary?
(a) Rs. 63960
(b) Rs. 78100
(c) Rs. 65300
(d) Rs. 89800


text:
# zbMATH — the first resource for mathematics

Trees and asymptotic expansions for fractional stochastic differential equations. (English) Zbl 1172.60017
Summary: We consider an $$n$$-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameter $$H>1/3$$. We derive an expansion for $$E[f(X_t)]$$ in terms of $$t$$, where $$X$$ denotes the solution to the SDE and $$f:\mathbb R^n\to\mathbb R$$ is a regular function. Comparing to F. Baudoin and L. Coutin [Stochastic Processes Appl. 117, No. 5, 550–574 (2007; Zbl 1119.60043)], where the same problem is studied, we provide an improvement in three different directions: we are able to consider equations with drift, we parametrize our expansion with trees, which makes it easier to use, and we obtain a sharp estimate of the remainder for the case $$H>1/2$$.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G17 Sample path properties 60H05 Stochastic integrals 60H07 Stochastic calculus of variations and the Malliavin calculus 60G15 Gaussian processes
Full Text:
##### References:
 [1] E. Alòs, O. Mazet and D. Nualart. Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001) 766-801. · Zbl 1015.60047 · doi:10.1214/aop/1008956692 [2] F. Baudoin and L. Coutin. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic. Process. Appl. 117 (2007) 550-574. · Zbl 1119.60043 · doi:10.1016/j.spa.2006.09.004 [3] G. Ben Arous. Flot et séries de Taylor stochastiques. Probab. Theory Related Fields 81 (1989) 29-77. · Zbl 0639.60062 · doi:10.1007/BF00343737 [4] C. Borell. On polynomial chaos and integrability. Probab. Math. Statist. 3 (1984) 191-203. · Zbl 0555.60008 [5] L. Coutin and Z. Qian. Stochastic rough path analysis and fractional Brownian motion. Probab. Theory Related Fields 122 (2002) 108-140. · Zbl 1047.60029 · doi:10.1007/s004400100158 [6] P. E. Kloeden and E. Platen. Numerical Solutions of Stochastic Differential Equations , 3rd edition. Springer, Berlin, 1999. · Zbl 0701.60054 [7] T. Lyons and Z. Qian. System Control and Rough Paths . Oxford Univ. Press, 2002. · Zbl 1029.93001 · doi:10.1093/acprof:oso/9780198506485.001.0001 [8] T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215-310. · Zbl 0923.34056 · doi:10.4171/RMI/240 · eudml:39555 [9] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140. · Zbl 1058.60037 · doi:10.1016/j.jfa.2004.01.002 [10] Y. Hu and D. Nualart. Differential equations driven by Hölder continuous functions of order greater than 1/2. Proceedings of Abel Symposium . To appear, 2007. · Zbl 1144.34038 · doi:10.1007/978-3-540-70847-6_17 [11] A. Neuenkirch. Reconstruction of fractional diffusions. In preparation, 2007. · Zbl 1141.60043 [12] A. Neuenkirch, I. Nourdin and S. Tindel. Delay equations driven by rough paths. Preprint, 2007. · Zbl 1190.60046 [13] I. Nourdin and T. Simon. On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion. Statist. Probab. Lett. 76 (2006) 907-912. · Zbl 1091.60008 · doi:10.1016/j.spl.2005.10.021 [14] I. Nourdin and T. Simon. Correcting Newton-Cotes integrals by Lévy areas. Bernoulli 13 (2007) 695-711. · Zbl 1132.60047 · doi:10.3150/07-BEJ6015 · euclid:bj/1186503483 [15] I. Nourdin and C. A. Tudor. Some linear fractional stochastic equations. Stochastics 78 (2006) 51-65. · Zbl 1102.60050 · doi:10.1080/17442500600688997 [16] D. Nualart. The Malliavin Calculus and Related Topics , 2nd edition. Springer, Berlin, 2006. · Zbl 1099.60003 [17] D. Nualart and A. Rǎşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81. · Zbl 1018.60057 · eudml:42846 [18] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Preprint, Barcelona, 2006. · Zbl 1169.60013 [19] V. Pipiras and M. S. Taqqu. Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 (2000) 251-291. · Zbl 0970.60058 · doi:10.1007/s004400000080 [20] E. Platen and W. Wagner. On a Taylor formula for a class of Itô processes. Probab. Math. Statist. 2 (1982) 37-51. · Zbl 0528.60053 [21] A. Rößler. Stochastic Taylor expansions for the expectation of functionals of diffusion processes. Stochastic Anal. Appl. 22 (2004) 1553-1576. · Zbl 1065.60068 · doi:10.1081/SAP-200029495 [22] A. Rößler. Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations. Stochastic Anal. Appl. 24 (2006) 97-134. · Zbl 1094.65008 · doi:10.1080/07362990500397699 [23] A. A. Ruzmaikina. Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100 (2000) 1049-1069. · Zbl 0970.60045 · doi:10.1023/A:1018754806993 [24] M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333-374. · Zbl 0918.60037 · doi:10.1007/s004400050171
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.