# Phase portrait

#### Phase portrait

Using an HSV color circle, this is what f(z)=z looks like.  x ′= x− 2y, y = 3x− 6y. A system’s phase portrait is the set of all trajectories that occur in its phase space. Phase Portraits of 1-D Autonomous Equations In each of the following problems -: (a) ﬁnd all equilibrium solutions; (b) determine whether each of the equilibrium solutions is stable, asymptotically stable or unstable; and (c) sketch the phase portrait. Starting from the. Phase portraits are an invaluable tool in studying dynamical systems. The phase portrait of the undamped second-order system 16. Why state plane analysis? 1. The transformation maps the (u, v) plane into the (x, y) plane preserving the qualitative features of the phase curve. I noticed that many hits to my blog, Phase Portrait , are from people searching for things like " phase portrait matlab ". 2018. To draw a phase portrait you will need more than just one solution. 5 Saddle phase portrait. They rely on the visual encoding of complex numbers: if z=reit then the rainbow colours represent the phase of z. MATLAB offers several plotting routines. An Interactive Applet powered by Sage and MathJax. It is very easy to generate a phase portrait like the one shown below. A phase portrait is a set of parameterized curves in the (x, v) plane that show the dynamics of the differential equation for a variety of initial conditions. (1) dx dt. As discussed earlier, we see that (±π, 0) are saddle point but now the origin has become a stable node. A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. 2. The arrows in the figure indicate the direction of increasing t . x˙1 = ¡x1 ¡2x2x2 1 +x2; x˙2 = ¡x1 ¡x2. m — plot Fourier sine series In the phase portrait of the system X 3 1 2 1 X the origin is a an A from AA 1 Phase Plotting Di erential Geometry Hyperbolic Geometry Conclusion Phase Portraits of Hyperbolic Geometry Scott B. Determine the critical points of the equation. In fact, analytic functions can be reconstructed based only on a phase portrait, up to a positive factor. js. Phase portraits for complex functions. ) For more details on these and related concepts, see State space - Scholarpedia. Phase Portraits of 1-D Autonomous Equations. $\endgroup$ – AnonSubmitter85 Feb 19 '14 at 10:45 add a comment | So I want to be able to draw the phase portrait for linear systems such as: I am completely confused, but this is what I have come up with so far: Step 1: Write out the system in the form of a matrix. indicting that the response of the system is constant in amplitude. Edit on desktop, mobile and cloud with any Wolfram Language product. Oct 21, 2005 Phase portrait. 14). In each of the following problems [ 1]-: (a) find all equilibrium solutions; (b) determine whether each of the  Aug 8, 2012 Fifty years ago, FitzHugh introduced a phase portrait that became famous for a twofold reason: it captured in a physiological way the qualitative  Apr 24, 2018 We propose a method to obtain phase portraits for stochastic systems. Another important tool for sketching the phase portrait is the following: an eigenvector for a real eigenvalue corresponds to a solution that is always on the ray from the origin in the direction of the eigenvector . Phase Portraits of Linear Systems Consider a linear homogeneous system . Phase portrait. Phase portrait of a nonlinear system numpy. The stable equilibria are at θ = 2nπ and the unstable ones at θ = (2n+1)π (n ∈ Z), by considering J = 0 1 −(g/l)cosθ 0 ! . The phaseportrait command provides an easy and clean way to plot direction fields and solution curves of 1- and 2-D systems. The vector field is made up of vectors in the direction of dy/dx at individual points (x,y). I found an interesting link that has some code and discussion on this topic. Enter two sequences that represent the population of rabbits and foxes. This sketch is called the phase portrait. Qualitative Behavior: Phase Portraits. The "quiver" function may be ideal to plot phase-plane portraits. The following phenomena can be seen: stable and unstable saddle points, lines of equilibria, nodes, improper nodes, spiral points, Phase Portraits of 2-D Linear Systems with Zero Eigenvalue For each of the following systems, • Find general solutions; • skecth the phase portrait; • determine whether the equilibrium (x,y) = (0,0) is stable or unstable; • determine whether the equilibrium (x,y) = (0,0) is asymptotically stable. newcastle. Dynamical regimes, such as a resting state or periodic oscillation, correspond to geometric objects, such as a point or a closed curve, in the phase space. 1 Answer. So here is the full phase portrait with some more trajectories sketched in. We define the equilibrium solution/point for a  Consider a systems of linear differential equations x′ = Ax. By hand, sketch typical solution curves in the xy -plane determined by the graphs of the equilibrium solutions. The eigenvectors are displayed both graphically and numerically. dy/dx = (2/π)y − sin y. phase portrait. Of course, the population of predators is related to the population of its prey. Discuss a way of obtaining a phase portrait of the equation. Illustration   May 9, 2015 We give an easy algorithm for obtaining the local phase portrait near the origin of a bi-dimensional differential system and we provide several  We have developed and implemented in the QPORTRAIT program a qualitative simulation based method to construct phase portraits for a significant class of  1. Phase portrait: Sketch trajectories. tan (x1 + x2) ODEintWarning: Illegal input detected (internal error). phase. You can vary any of the variables in the matrix to generate the solutions for stable and unstable systems. The set of all trajectories is called phase portrait. 2018 Aug;21(11):645-653. Phase Portrait. The behavior of x 2 versus t is similar. Let $A$ be a $2 \times 2$ matrix. Positive real part. 1) For a linear system, you just need to find the eigenvalues of matrix A and the corresponding eigenvectors. time (3 ODEs) $\begingroup$ Perhaps you can describe how to make a phase portrait and someone here will help you implement it in Matlab or Mathematica. For each of the following systems, follow steps similar to those in Exercise 21 to locate the equilibrium point of the system, classify, the equilibrium, and sketch the phase portrait. pplane is a great program that makes it very easy to tackle nonlinear time-invariant differential equations of two states. A trajectory is the directed path traced out by a solution. Русский: Фазовый портрет системы с особой точкой типа  Constructing phase portrait. neurodyn » Phase Portrait. The following statements define evenly spaced initial conditions of the form (x 0, 0). Hence, the phase portrait is that of the node in. Step 3: Using the eigenvectors draw the eigenlines. Phase Portrait (2 ODEs) Plots the phase portrait of a pair of differential equations. (2) . The equilibrium point is called a node if and only if it is neither a source nor a sink. Sketch the phase portrait of the original system by translating that of the transformed system. Maple - Phase Portraits. For the one dimensional system studied above, the phase space is two dimensional: qgives one dimension, dfield (direction field) and pplane (phase plane) are software programs for the interactive analysis of ordinary differential equations (ODE). Notation: Determinant of A: Trace of A: When the initial conditions are different, the phase trajectories are a family of ellipses starting from the point The phase portrait is shown in Fig. For the system (1) dx dt = 2 -4 x-15 y (2) dy dt = 4 -x2 we will identify the critical points,and then plot several trajectories and the related slope field, by utilizing Maple's plots, plottools and DEtools packages. It turns out that the phase portrait for this case is the same as the one depicted in Figure 1. The generic form of what you will type will look like this: The generic form of what you will type will look like this: The direction field solver knows about trigonometric, logarithmic and exponential functions, but multiplication and evaluation must be entered explicitly ( 2*x and sin(x), not 2x and sin x ). - jithinbp/phase-portrait Making Phase Portraits. A phase portrait is a plot of multiple phase curves corresponding to different initial conditions in the same phase plane (Tabor 1989, p. The two dimensional case is specially relevant, because it is simple enough to give us lots of information just by plotting it. This is consistent with the picture. This is the classical time series behaviour observed when a system is critically damped. Solver for the Lorenz equations (solver_Lorenz_MC. Figure 3. Phase Portrait and plots of variables vs. lrn takes four possible values: 'l' for left arrow, 'r' for right arrow, 'n' for node or equilibrium point, and 'u' for undefined value of the derivative. For the linear . Lindstrom and Paul Vrbik CARMA University of Newcastle Jonathan M. a path always tangent to the vectors) is a phase path.  dy dt = sin(πy). Indicate direction of motion by arrows point-ing in the direction of increasing t Direction of Motion on Half Line Trajectories: • If λ1 > 0 then x(t) = c1eλ1tv1 – moves out to ∞ for t → ∞ (outwards arrow on H1+) – approaches 0 for t → −∞ • If λ1 < 0 then x(t) = c1eλ1tv1 – approaches 0 for t → ∞ The phase portrait along with several graphs of x 1 versus t are given below. Evolution of a dynamical system corresponds to a trajectory (or an orbit) in the phase space. PhasePlane(sys,tspan,icond) plots the the phase plane portrait for a general second order nonlinear system defined by the function sys(t,x). It is best to draw the phase portrait in small pieces. warnings. The nature of the equilibria can be determined from the eigenvalues of the matrix 1 1 4 y x The rows are the partial derivatives with respect to and of the system. Solution for systems of linear ordinary differential equations - Phase portraits - Duration: 59:19. 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). warn (warning_msg, ODEintWarning) RuntimeWarning: divide by zero encountered in double_scalars x2_d = x1 - 4 * 1/np. 3. Hello. Together we will look at how to classify critical points, or Equilibrium Solutions, and their graphs based on Eigenvalues and Eigenvectors: Saddle, Nodal Source or Sink, Degenerate or Improper Nodal Source or Sink, Center or Spiral Source or Sink, and Stable and Unstable Saddle points. They consist of a plot of typical trajectories in the state space. Phase portraits of a system of ODEs. Phase plane. Trajectories:. Different initial states result in different trajectories. field, a phase portrait is a graphical tool to visualize how the solutions of a given system of differential equations would behave in the long run. The equilibrium point is called a source if and only if any solution y ( t) to the autonomous equation such that y (0) is (or not) close to , then y ( t) will move away from when . 5) has θ¨= − g l sinθ. Phase Portrait of Lotka-Volterra Equation. We think of this as describing the motion of a point in the plane (which in this context is called the phase plane), with the independent variable as time. Your browser The phase portrait shares characteristics with that of a node. e. aPicon are tokens that will be placed on the phase line when the due date is past. Phase portraits are a technique for visualising complex valued functions of a single complex variable. mw. Any solution that starts near the origin will tend towards the origin as . By choosing an initial vector $\ mathbf{x}_0 = \begin{bmatrix}x_0 \\ , we can obtain the Abstract. Includes lots of MATLAB and LaTeX (computer typesetting) tips along with commentary on all things engineering and some things not. Each set of initial conditions is represented by a different curve, or point. Using Matlab to get Phase Portraits. The originis arepeller, hence unstable. In physical systems subject to disturbances, the Phase portrait: Sketch trajectories. In this case these are the equations for a predator-prey model: Using Matlab to draw phase portraits This is a quick notes to help you draw phase portraits using the quiver command in Matlab. Abstract-A thorough understanding of the phase portrait, or more precisely the stability region and the stability boundary of a nonlinear dynamic system is the Jan 11, 2019 Thus, we have the phase portrait in Figure 3. Complex eigenvalues, w/ Non-zero real part UNSTABLE If real part is positive ASYMPTOTICALLY STABLE If real part is negative. I tried the following code, which I am pretty sure is incorrect, since it doesn't update the t for Q(t) after the initial step. The phase portrait is a representative sampling of trajectories of the system. Q Is it possible to plot 3D phase portraits? A Yes: there are several ways to go. Plots of variables vs. time (2 ODEs) Plots the phase portrait of a pair of differential equations and plots the dependent variables as a function of t (the independent variable) on separate axes. Solutions x for which Ax = 0 correspond to equilibrium solutions, and are called critical points. Q-25-10-240 (single-phase) Single phase for 1. Phase portraits for Equilibrium Solution, Phase Portrait. Phase portraits for simple harmonic motion. Two distinct real eigenvalues, opposite signs Always UNSTABLE Spiral Point. Unfortunately for them, they This page plots a system of differential equations of the form dx/dt = f(x,y), dy/dt = g(x,y). In the phase portrait above we are plotting the angular position state against the angular velocity state. Maple and Phase Portraits. I want to plot a phase portrait for a system of differential equations of the form: xdot= f(t,x,y) ydot= g(t,x,y) where t is the independent variable. I've been asked to do the following and now I'm struggling to do part (c) (Part (a) and (b) are irrelevant!) code for part(c): Qmaple. Borwein Commemorative Conference, September 2016 Last Revised September 26, 2017 https://carma. au/scott/ I am hoping to draw a phase portrait (plane) of this system using Python, but all of the tutorials I could find online were only for not-time-dependent systems. m — phase portrait of 2D ordinary differential equation phaseg. A phase portrait, in it’s simplest terms, is when we plot one state of the system against another state of the system. 2 Answers. Usually phase portraits only include the trajectories of the solutions and not any vectors. "Phase portrait of the high-dimensional olfactory dynamics is shown in two representative dimensions Feb 12, 2019 I noticed that many hits to my blog, Phase Portrait, are from people searching for things like "phase portrait matlab". Now, it will be easier to explain the remainder of the phase portrait if we actually have one in front of us. 3. The system we shall consider is. here is our definition of the differential equations: f = @(t,Y) [Y(2); -sin(Y(1))]; Phase plane portrait is a very important tool to study the behavior and stability of a non-linear system. This Demonstration shows a phase portrait of the Lotka–Volterra equations, including the critical points. Autonomous systems. 3D Phase Portrait with direction fields and sample solution curves [duplicate] I have seen NDSolve and ParamtericPlot3D used for things like the Lorenz Attractor, but was wondering if there is a way to draw a 3D phase portrait for a system. Let's make this more concrete: Say I want to draw a nice phase portrait for$\dot{x } = -y\dot{y} = -x$First I generate the vector field: var('x y') Mar 5, 2015 English: Phase portrait of the system with a stable focus equilibrium point. doi: 10. Phase Portraits I've noticed lately that I've been getting a lot of search hits here and at the home page about plotting phase portraits (in MATLAB). The software is described in detail in the manual Ordinary Differential Equations using MATLAB. In particular, phase portraits for such systems can be classified according to types of eigenvalues which appear (the sign of the real part, zero or nonzero imaginary parts) and the dimensions of the generalized eigenspaces. m) aPicon are tokens that will be placed on the phase line when the due date is past. A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Tracing a path of vectors yields a solution to the ordinary differential equation at a set of initial conditions. Just define the differential equations. edu. m) Lorenz Equations. For the system. m — phase portrait plus movie of pendulum motion Computational subroutines o3. (The domain in the portraits phase portrait. Indicate direction of motion by arrows point- ing in the direction of increasing t. This geometric feature is typical of systems whose coefficient matrices have complex eigenvalues. Phase Line. That makes the origin an attractor , and hence, all solutions will be drawn into the origin, making the system asymptotically stable . 1080/10255842. (In practice, one draws representative ones to depict all qualitatively different possibilities. A phase portrait for the simple harmonic oscillator Phase Plane Portraits. / #0 1$ - 1 ) 1 ) #0 1$- 1 ) 1 ) In the procedure- isthe nameof the -ﬁle deﬁningthe equations, is the time interval wanted for the solutions, , is of the form #$ and deﬁnes the plotting window in the phase plane, and A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. PHASE PORTRAITS OF LINEAR SYSTEMS 3 2. here is our definition of the differential equations: To generate the phase portrait, we need to compute the derivatives y′1 and y′2 at t=0 on a grid Once we have called quiver, we overlay the phase plane portraits of solutions obtained by specifying different initial conditions -- notice how the solutions follow the arrows of the quiver plot, then get attracted to the same limit cycle (regardless of whether we start inside or outside of the cycle). If the motion is outward, while if it is inward. The phase portrait is as follows: 79. A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase space (q;p). Phase Portraits, Eigenvectors, and Eigenvalues. (2) has a saddle point at (0,0) The first thing you need when you want to solve a system of differential equations in Matlab is a function that computes the derivative (i. So, the nature of equilibrium point is determined by the roots of this polynomial. 5. We may generate the phase portrait of a system of nonlinear first order DEs using Maple. The geometric properties of the phase portrait are closely related to the algebraic characteristics of eigenvalues of the matrix A. = 2 - 4 x - 15 y. Fokker-Planck equation, we separate the dynamics  corresponding orbit is simply a point on the phase plane with coordinates (x, y). Learn more about 3-d phase portraits, ode. 1. The following worksheet is designed to analyse the nature of the critical point (when ) and solutions of the linear system X'=AX. matsciencechannel 5,707 views Phase Portraits, Eigenvectors, and Eigenvalues. Indeed, the important feature of this phase portrait is the spiraling nature of trajectories approaching the origin. Classify each critical point as asymptotically stable, unstable, or semi-stable. Its syntax  Aug 28, 2018 3d phase portrait. With only one eigenvector, it is a degenerated-looking node that is a cross between a node and a spiral point (see case 4 below). [′fāz ‚pȯr·trət] (control systems) A graph showing the time derivative of a system's position (or some other quantity characterizing the system) as a function of position for various values of initial conditions. If you want to see how one sequence affects another sequence, you can graph a phase plot on your TI-84 Plus calculator to represent the data. In the sequence for […] This article describes how to draw phase portraits for two classic differential equations: the equations of motion for the simple harmonic oscillator and the pendulum. An autonomous system is a system of ordinary differential equations which does not explicitly depend on the independent variable. • Phase plane analysis of linear systems. Phase portrait are geometric representations of the trajectories of a dynamical system in the phase plane, and are an important tool in the study of dynamical systems . The task of our investigation is to find out all the different (in the topological sense ) phase portraits in a Poincare circle and indicate the coefficient criteria of their  Consider the autonomous DE. $\endgroup$ – AnonSubmitter85 Feb 19 '14 at 10:45 add a comment | The x, y plane is called the phase y plane (because a point in it represents the state or phase of a system). Here the damping is so large that the trajectories approach the origin as stable node, which means that the system will overshoot the origin at max once and then settle at the stable node. m — phase portrait plus graph of second order ordinary differential equation phasem. which can be written in matrix form as X'=AX, where A is the coefficients matrix. the right hand sides of the differential equations). The eigenvalues at the critical points are also calculated, and the stability of the system with respect to the varying parameters is characterized. Let y = θ˙, so that θ˙ = y, y˙ = − g l sinθ. For any system that obeys the markov property we can construct such a diagram, with one dimension for each of the system’s stocks. Saddle Point. Loss of control accidents, which lead to thousands of deaths every year in America alone, are often caused by a miscalculated action of the driver or a  Phase portrait Potential energy and phase portrait of a simple pendulum.  dy dt = −(y − 1)(y − 3)2. A phase portrait for a two-dimensional autonomous system of ordinary differential equations can be created using the StreamPlot function. Red indicates a phase of 0 and the rainbow goes through red, yellow, green, blue, Phase Portraits¶ In this notebook we’ll look at how to generate phase portraits. m) Phase portrait (3D systems) Computation of the phase portrait for 3D systems (phase_portrait_3D. We will see that the qualitative nature of the solutions is determined by the eigenvalues of the coefficient matrix. If we were graphing a vector field in the phase plane of the linear system Y / =. Phase Portrait Personal weblog of Ted Pavlic. Direction of Motion on Half Line. The main tool will be phase portraits, which are sketches of the trajectories of solutions in the xy-plane (now called the phase plane). FIG. Then, we will look at seven examples together. Source code: phase, waveform, and plotarea Built with Processing and Processing. Sinceweareinterested inthosecaseswithnonzerorealpart,therearetwopossibilities: positive or negative realpart. • Limit cycles. The expression: is called characteristic polynomial. Example: a simple pendulum of length l (as in §2. Phase spaces are used to analyze autonomous differential equations. Epub 2018 Oct 27. NEW: See Phase Portraits, a new blog post discussing dealing with nonlinear differential equations in MATLAB, for more information about alteratives to pplane in MATLAB. (with t as the  A phase portrait is a plot of multiple phase curves corresponding to different initial conditions in the same phase plane (Tabor 1989, p. Note that the x-axis, being angular, wraps onto itself after every 2π radians. These techniques, which  Stable and unstable manifolds of equilibrium points and periodic orbits are important objects in phase portraits. At , the eigenvalues are and and at , there is one positive eigenvalue and one negative eigenvalue. Phase Portraits and Vector Fields of Systems. Each set of initial conditions is represented by a  Oct 29, 2018 In this section we will give a brief introduction to the phase plane and phase portraits. In order to properly model the heartbeat cycle we must assume that the mathematical equations be drawn from a particular class of equations that will adequately describe the fundamental features of heart physiology. 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. Red indicates a phase of 0 and the rainbow goes through red, yellow, green, blue, Maple and Phase Portraits We may generate the phase portrait of a system of nonlinear first order DEs using Maple. If 0 < r 2 < r 1, then the trajectories will have the same pattern as in figure (a) below, but the direction will be away from the critical point at the origin. Run with full_output = 1 to get quantitative information. In this case the critical point is again called a node or a nodal source. representing the general picture are called the phase portrait. MathQuest: Differential Equations. In this case these are the equations for  In this video David explains how a phase constant can be used in order to shift the graph of an oscillator left or right. Here is a primitive procedure in a ﬁle that provides this easily: - , , # $, . For the general case, where A A has eigenvalues λ1<0<λ2, Oct 27, 2017 The great advantage is that for one-dimensional systems it is easy to construct the phase space portraits, and thus extract information and Nov 20, 2017 Hi there,. • Phase plane analysis of nonlinear systems. The flows in the vector field indicate the time-evolution of the system the differential equation describes. Phase portrait (2D systems) Computation of the phase portrait for 2D planar systems (phase_plane. 5: Find the critical points and phase portrait of the given autonomous first-order DE. Python-pygame based app to simulate three diemsional phase portraits. 1. Comput Methods Biomech Biomed Engin. (By Thomas Scofield) View PNG (It opens in a new tab, so you may need to allow popups). share | cite | improve this answer answered Feb 9 '16 at 22:18 Phase portrait of a nonlinear system numpy. Trajectories in these cases always emerge from (or move into) the origin in a direction that is parallel to the eigenvector. 3m in on the origin of a linear system, the phase portrait will look exactly the same. COMPLEX EIGENVALUES (WITH NONZERO IMAGINARY PART) Thesecomeincomplex-conjugatepairs,sobotheigenvalueshavethesamerealpart. The state is a single number or a set of numbers (a vector) that uniquely defines the properties of the dynamics of the system. Example. The entire field is the phase portrait, a particular path taken along a flow line (i. Lecture 4: Phase Portraits (Phase Plots) The dynamic properties of a particle are described by the state of the system. m — set up for Fourier sine series fsine. Node Two distinct real eigenvalues, same sign UNSTABLE If both eigenvalues are positive ASYMPTOTICALLY STABLE If both are negative. Thus, the equilibrium x = 0 is a saddle, hence unstable, when = 0. Phase space is the underlying space of the system’s variables. Labelled Axes, Phase Portrait. The solution is on the ray in the opposite direction. Consider the homogeneous linear first-order system differential equations x'=ax+by y'=cx+dy. 19, 20 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. In this context, the Cartesian plane where the phase portrait resides is called Sketch the phase plane portrait of a 2D system of first order differential equations. Step 2: Find the eigenvalues and eigenvectors for the matrix. (1) has an unstable focus at (0,0). m — a simple 3D differential equation fset. Remember that a phase portrait is a collection of phase trajectories which differ by their initial conditions. Phase portraits can be treated as images and lend themselves well to a variety of image analysis and pattern recognition techniques. 05m portrait module pitch Connector pitch is 1. . dy y3 -6y2 +8y dx Figure 1: Phase potrait of under damped 2 Next we consider the over damped case. 1504215. So, in the following phase portraits of the linearizations, the ranges on the axis are from 1 to 1.$\begingroup\$ Perhaps you can describe how to make a phase portrait and someone here will help you implement it in Matlab or Mathematica. Its phase portrait is a representative set of its solutions, plotted as parametric curves. A phase diagram shows the trajectories that a dynamical system can take through its phase space. Phase Plane Plotter. In the past, I have advised people to look into trying pplane (and dfield) . For a much more sophisticated phase plane plotter, see the MATLAB  Nov 14, 2017 Concepts of phase plane analysis Phase plane, phase trajectory and phase portrait :- the second-order system by the following ordinary  Phase portrait. In our case the class from which we choose must have two degrees of freedom. We assume A is nonsingular,   Phase portraits can be treated as images and lend themselves well to a variety of image analysis and pattern recognition techniques.  dP dt = P(P2 − 1)(P − 3). Trajectories for the critical damping case. The following phenomena can be seen: stable and unstable saddle points, lines of equilibria, nodes, improper nodes, spiral points, Linear Phase Portraits: Matrix Entry 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. These techniques, which we briefly describe next, include Hough transforms, 12 Hausdorff measures, 17 artificial neural networks, 18 Kernel methods, and support vector machines. phase portrait

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