# Phase Portrait Plotter Matrix

Nonlinear Methods for the Social Sciences Stephen J. Real part of is positive and the FP is unstable. Hence, the phase portrait is that of the node in. A has a repeated eigenvalue with only one linearly independent eigenvector. The phase portrait is a representative sampling of trajectories of the system. In GPoM: Generalized Polynomial Modelling Pre-processing for global modelling. 1 The Phase Plane Example 2. A fixed point solution is a point in a phase portrait. Introduction Phase Plane Qualitative Behavior of Linear Systems Local Behavior of Nonlinear Systems Isocline Method I The algorithm of constructing the phase portrait by isocline method: 1. (ii) Some of the plots show direction ﬁelds of equations all whose solutions are increasing functions (at. Construction of the exponential matrix, including Jordan canonical form for 2 x 2 matrices. This observation will be very useful when we sketch phase portraits. The following plots have been produced with octave using the above procedure:. This work generates the solution The general solution is The first picture shows the direction field; the second shows the phase portrait, with some typical solution curves. The geometric properties of the phase portrait are closely related to the algebraic characteristics of eigenvalues of the matrix A. The initial conditions in the routine are placed on a circle with radius r. In order to obtain a general first overview over the different types of bifurcations that occur in the system, it is useful to plot phase portraits for different values of the control parameters and to compare them to each other. unstable manifolds of saddle points. m Matlab m-file for portraying an IFS based on a matrix of values henon. Has anyone out there used Igor to make a phase portrait (aka phase space plot) of an ODE? I attach an example (middle plot) below. We write ~x˙ = F~(~x) = ˆ x˙1 = x2, x˙2 = −sin(x1) The ﬁxed points are where F~(~x) = 0, which we write as F~(~¯x) = 0 for. It doesn’t have the simple point and click features of the MatLab pplane8program, but it is not very diﬃcult to use. Departmental approval. fitzhugh_nagumo. For math, science, nutrition, history. x L 1 y L 2 7. Phase portrait of the nonlinear system. PortraitDensity determines the density of the phase portrait. Plot 3D phase portrait. A fixed point solution is a point in a phase portrait. With the linearized equations, we now have U(x) = − R x x0 ζdζ = − 2+ 0 2. The last matrix says that , so. 3 in Differential Equations with MATLAB. In this study, L. which can be written in matrix form as X'=AX, where A is the coefficients matrix. Moreover, local phase portrait of a hyperbolic equilibrium of a nonlinear system is equivalent to that of its linearization. If necessary you will find the commands at the end of the Maple version of this example. We start by. This observation will be very useful when we sketch phase portraits. Classify differential equations according to their type and order. Histopathological assessment of lymph node metastases (LNM) depends on subjective analysis of cellular morphology with inter-/intraobserver variability. The left plot is a temporal representation of the system's development, with time $$t$$ being represented on the horizontal axis. PhasePlane(sys,tspan,icond) plots the the phase plane portrait for a general second order nonlinear system defined by the function sys(t,x). Sketch the phase portrait of this system. Hyperbolic equilibria are robust(i. On this page I explain how to use Matlab to draw phase portraits for the the two linear systems. The user must provide a field (matrix) of complex numbers z which covers the domain (typically a rectangle, a disk or an annulus) of the function, and a field of the same size with the corresponding values w=f(z). 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. In this section we will give a brief introduction to the phase plane and phase portraits. The graphing window at right displays a few trajectories of the linear system x' = Ax. The resulting gallery is shown in the two pages of phase plane portraits that comprise Figure 5. Mathgrapher contains many demonstrations covering most of the things you can do with it. This observation will be very useful when we sketch phase portraits. The family of all trajectories or solution curves (which started by different initial points) is called phase portrait. ±2 ±1 0 1 2 J 2 R We will show the repeller and attractor are the eigendirections of the matrix. 7c, we plot SLP and SIC metrics for the Bering Sea. A Windows based mathematical graphing tool for 2D and 3D Functions and Data, shaded surfaces, contour plots. Non-Linear Dynamics Homework Solutions Week 4: Strogatz Portion February 3, 2009 6. • Used MATLAB to sketch phase portraits and approximated changing behavior of dynamic systems in the neighborhood of the fixed point, then identified types of bifurcation • Nondimensionalized. Here are some of the principles of trajectory sketching:. The matrix plot enables the realization that chaos only occurs under selective amplitudes and periods of the perturbation. For math, science, nutrition, history. The phase. In this section we will solve systems of two linear differential equations in which the eigenvalues are real repeated (double in this case) numbers. 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. 12 Procedure to draw phase portrait in XY plane(2nd order) Find critical points: Eg: x=4x-3y ,y=6x-7y have critical point at (0,0) construct a phase plot (y vs x) find eigen values and eigen vector of the system equation eigen values are (-5 and 2) and corresponding eigen vectors are [1;3] and [3;2] and draw corresponding vector axes if eigen. Analysis tools include power spectrum calculation and Poincare sections. On the back of this guide is a flow chart which describes the process. Introduction. Sketching Non-linear Systems OCW 18. Show that the equilibrium point in phase space is a center, which means that the cylinder oscillates, and find the oscillation period of the cylinder. 552 Phase Plane Methods were used, then 50 solution curve segments have already been entered onto the graphic! Threaded orbits are added to show what happens to solutions that are plotted on longer and longer t-intervals. Hence, the eigenvalues of the matrix. Exercises and Examples 1. A phase portrait is a plot of a segment’s angular displacement versus its angular velocity (Barela et al. To display the graph of a function we choose some interval of values for , and plot the points of the graph whose -coordinates lie in the given interval. The software is described in detail in the manual Ordinary Differential Equations using MATLAB. In the case of centers and spirals you may also be asked to. 7c, we plot SLP and SIC metrics for the Bering Sea. Left plot is for a= 1 and right plot is for a= 1. In this section we will solve systems of two linear differential equations in which the eigenvalues are real repeated (double in this case) numbers. FitzHugh-Nagumo: Phase plane and bifurcation analysis¶ Book chapters. Here are some of the principles of trajectory sketching:. The system of equations is written u' = AA*u, where AAis a given 2×2-matrix anduis a column vector. On the plot at the bottom of the previous page, in the box at the right, draw a phase portrait near the xed point N A = K=4, N 0 = K=4. This box represents a blow-up of the box in the computer-generated plot at the left. * Updated many function names to be more consistent with python coding. Mosteller and Tukey’s World War II example Simpson’s Paradox. If is a function, then the \emph{graph} of this function is the collection of points in the plane, where is the domain of. 3 in Differential Equations with MATLAB. JPlotter JPlotter is an open source math plotter that can draw graphs of arbitrary mathematical functions. Phase Plane Plotter. In this instance, numerical integration will usually be the only way forward. 3 The Phase Plane for a Conservative System A second-order diﬀerential equation for a variable x(t) can always be converted to two. Obtain a Describing Function for this nonlinear element. Show that the dynamical system associated to the motion of the cylinder is linear and find its matrix. Problem set 5: Solutions Math 207A, Fall 2018 1. Just as we did for linear systems, we want to look at the trajectories of the system. 7-26, Section 6 Ex. Th: Phase Portraits and Lotka-Volterra Equations For plotting the population data please use the code plot_population. Be able to write a system of linear (algebraic) equations as a matrix equation. You may integrate and analyse systems of up to 20 coupled ordinary differential equations (ODE's). We show by treating a concrete example how you can use Matlab to plot the phase portrait of a linear system in the plane. Then we will analyze the geometric effect of transforming a linear map to its Jordan Normal Form. In this case, altering the output color they emit (blue outside and green inside the mesh). Differential equation. 1 x x 1 2 v 2 v 1-1 1-1 Phase portraits for 2 × 2 systems. So I want to be able to draw the phase portrait for linear systems such as: x'=x-2y y'=3x-4y 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. Finding Stationary Points This guide describes how to use the first and the second derivatives of a function to help you to locate and classify any stationary points the function may have. The initial conditions in the routine are placed on a circle with radius r. Finally, we show how to generate a very naive 'phase portrait' of trajectories for different values of the model parameter eps. Perhaps its existence is unknown or its effect unsuspected. This observation will be very useful when we sketch phase portraits. Sketch the phase plane portrait of a 2D system of first order differential equations. Povinelli Department of Electrical and. Phase spaces are used to analyze autonomous differential equations. In this discussion, we will only plot solutions of autonomous, first order differential equations; that is, equations of the form To begin, type pline. The following problems discuss Lyapunov stability in the case when the Lyapunov theorem is non-applicable. In order to obtain a general first overview over the different types of bifurcations that occur in the system, it is useful to plot phase portraits for different values of the control parameters and to compare them to each other. On this page I explain how to use Matlab to draw phase portraits for the the two linear systems. x L 1 y L 2 7. We will plot the derivatives as a vector at each (y1, y2) which will show us the initial direction from each point. See Figure 11, we sketch phase portraits for the case when A has two eigenvalues ‚1 > ‚2 > 0. We compute the rank by computing the number of singular values of the matrix that are greater than zero, within a prescribed tolerance. 0 The fixed point in 0, 0 is unstable ----- Check. Duan in his recent published book [1, §5. Here are some general facts to remember when analyzing phase portraits of linear maps: If the matrix (A - I) is nonsingular then the origin is the only fixed point. Nonlinear Systems and Phenomena Application 9. Plot the curve S(x) = in state-space (phase plane) 2. A Direction Fields and Phase Portraits. I have to plot its phase portrait using this code: And if first line was 9x - 21y - 10z + 5 where would 5 goes in the matrix? matlab. Geometric structures occur in this processing space that are called trajectories or attractors. See examples/geneticswitch. The Scientific World Journal is a peer-reviewed, Open Access journal that publishes original research, reviews, and clinical studies covering a wide range of subjects in science, technology, and medicine. Use interactive calculators to plot and graph functions. I could imagine myself, just as Strogatz, trying to see if a phase portrait of a reversible system resembles something in nature, such as a manta ray (as in one of the examples he worked out). The Jacobian matrix is A = 0 1 3x2. The phase space plot and such a family of trajectories together are a phase space portrait, phase portrait, or phase diagram. Stability Analysis for ODEs Marc R. c) Write the Matlab code to plot the phase portraits. Step 3: Using the eigenvectors draw the eigenlines. This is a practicing course for MATLAB taught by Ahmed Rezk and Ahmed Mahdy. The dimension of the phase space is the number of intial conditions required to uniquely specify a trajectory; it is the number of variables in the dynamical system. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. This is a topic that’s not always taught in a differential equations class but in case you’re in a course where it is taught we should cover it so that you are prepared for it. 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. For all ﬁgures in this article, the plot range of x and y from known densities is set to include 99% quantiles of the target density. Essentially, you only need to solve the differential equations and then plot the result. A (discrete) phase portrait of the system ~x(t+ 1) = A~x(t) shows discrete trajectories for vari-ous initial states, capturing all the qualitatively diﬀerent scenarios (as in Figure 6). That makes the origin an attractor, and hence, all solutions will be drawn into the origin, making the system asymptotically stable. In this section, we will show you how to plot data, modify plots and save your work. It does so by examining symmetric matrices, the damped harmonic oscillator, and other normal matrices. Python classes. Using functional magnetic resonance imaging (fMRI), we studied the neural correlates of the complexity of rhythmic ﬁnger tapping. Although the literature on nonlinear methods is vast, most of it has been written for applications that do not share the concerns or intellectual traditions of the. 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. How can I draw phase portrait of a matrix with given several initial points? what the phase portrait of a matrix is and you want to plot them together. If the motion is outward, while if it is inward. The matrix plot enables the realization that chaos only occurs under selective amplitudes and periods of the perturbation. Phase spaces are used to analyze autonomous differential equations. Plotting graphs of functions. The Van der Pol equation has been illustrated as an example. Using Matlab to get Phase Portraits Once upon a time if you wanted to use the computer to study continuous dynamical systems you had to learn a lot about numerical methods. m Matlab m-file for portraying an IFS based on a matrix of values henon. Analysis tools include power spectrum calculation and Poincare sections. (Leave out the special case when one of the eigenvalues is 1. By creating phase plane diagrams of our system we can visualize these features, such as convergence, equi-. Answers [1] General solutions: x y = C1 2 1 +C2e−5t 1 3. Phase space plots are difficult to draw, since motion must be built into the plot. Second order nonlinear ODEs. Prove that λ is a generalized eigenvalue of the matrix pair K, M if and only if it is an ordinary eigenvalue of the matrix P = M-1 K. One particular function called DEplot provides very attractive phase portraits. 0 : Return to Main Page. Draw small line with slope. We ﬂnd these. unstable manifolds of saddle points. In the adaption phase, it adjusts neighborhood neuron pair's connected/disconnected topology constantly according to the statistics of input feature data. Let matrix A = 1 0 0 2 3 1 4 1 0 Then A looks triangular (upper bandwidth of 0). Moreover, local phase portrait of a hyperbolic equilibrium of a nonlinear system is equivalent to that of its linearization. Phase portraits (by using solutions) of the. In practice, the width is obtained by using an image. However, we can view this dynamically in MATLAB using the program pline. Phase Portraits and Equilibrium As discussed many times in the previous sections, there are three general solution types, Numerical, Analytic, and Qualitative. Handles MIMO systems by allowing choice of input and output. Week 5 : Chapter 1 Cont. Your browser. Closed-loop frequency response-Constant gain and phase loci, Nichol's chart and their use in stability study of systems. Show that y = C sin(t)+ D cos(t) is a family of solutions of y00+y = 0, and ﬁnd all solutions that satisfy the following constraints. Example Consider the. , the stable and unstable points where ) regulate the portrait of the phase plane. The Simulink. The journal is divided into 81 subject areas. Posts about Phase Portraits written by tsvhh Computing packages partition matrix pdf PDF-XChange Viewer pdfsync pdfTeX PGF Phase Portraits Plot poster PracTeX. Phase portrait. Note that the right-hand expressions are not continuously differentiable at $(0,0)$ and there is no linearization at this point, which explains that while stable, the point $(0,0)$ does not look (after close examination) like a standard node. Then, we obtain the phase portrait of the projected reduced system simply by changing the direction of time on S r (p) in the phase portrait of. Let’s use the following two-dimensional system as an example. So far, we have looked at a scalar equation for a membrane with a single nonlinear channel. 3 Theorem (The Fundamental Theorem For Linear Systems) [12] Let be an matrix. Properties of flows and orbits. Alex, assuming that you are talking about a u-w (position-velocity, sort of) phase plot, here you go. 2: Phase curves in the vicinity of centers and saddles. Using Matlab for Autonomous Systems. The phase portrait on the sphere is equivalent to that on the plane (x, y). This page intentionally left blank Economic Dynamics Phase Diagrams and Their Economic Application Second Edition This is the substantially revised and restructured second edition of Ron Shone’s successful undergraduate and graduate textbook Economic Dynamics. Bode plots, Polar plots, Log-magnitude Vs phase plots, Nyquist stability criterion, Stability analysis, Relative stability, Gain margin, Phase margin, Stability analysis of system using Bode plots. A plot of position and velocity variables as a function of time is sometimes called a phase diagram. The phase portraits is able to perfectly capture all of the nonlinear trajectories and display them in a way that would be otherwise difficult. Using Matlab to draw phase portraits This is a quick notes to help you draw phase portraits using the quiver command in Matlab. Scalar, linear higher order differential equations: relation with systems of differential equations. This video is unavailable. The network of representative neurons, first create the necessary neurons according to the local density of the input data in the growth phase. Control System - This Video tutorial is meant to provide the readers the know how to analyze the control systems with the help of mathematical models. PhasePlane(sys,tspan,icond) plots the the phase plane portrait for a general second order nonlinear system defined by the function sys(t,x). Properties of flows and orbits. Depending on various factors, different trajectories can evolve for the same system. PhasePlane(sys,tspan,icond) plots the the phase plane portrait for a general second order nonlinear system defined by the function sys(t,x). Try all of the exercises. Hence, the two eigenvalues are negative. B) The same mesh loaded into a BSim environment. Sketch the phase plane portrait of a 2D system of first order differential equations. To get started, plot your data and open the Figure Palette from the Figure View menu. Remove all;. A phase portrait is a plot of multiple phase curves corresponding to different initial conditions in the same phase plane (Tabor 1989, p. Exercises and Examples 1. Right panels: phase space. Perhaps its existence is unknown or its effect unsuspected. In particular, plot -T svg, plot -T ai, plot -T ps, plot -T cgm, plot -T fig, plot -T pcl, and plot -T hpgl are affected by the environment variable PAGESIZE. Printing xs over it give you the trajectory for the initial conditions you have chosen. This will include deriving a second linearly independent solution that we will need to form the general solution to the system. First define the right hand side function f of the differential equation as. 552 Phase Plane Methods were used, then 50 solution curve segments have already been entered onto the graphic! Threaded orbits are added to show what happens to solutions that are plotted on longer and longer t-intervals. PortraitDensity determines the density of the phase portrait. The phase portrait is a spiral which at the point (0;1) is directed downward since c= 1 < 0. If the motion is outward, while if it is inward. However, we can see the trajectories at infinity on the sphere, these are the trajectories moving along the equator, i. The following worksheet is designed to analyse the nature of the critical point (when ) and solutions of the linear system X'. See its phase portrait on the next. Here is an example of a plot of a sine wave. Graphing Differential Equations. A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Mathematical concepts that are taught include how to plot data points from a table, how to label ordered pairs, and how to name coordinates on a grid. The graphical representation of the solutions is often referred to as a phase portrait. Second order nonlinear ODEs. 12 For this case, a Q estimate (using the distribution of points along the major axis of the phase portrait) was used to monitor OSNR, while the width of the phase portrait was used to monitor CD. Determined appropriate matrix values for Q and R to achieve a specific bank angle (phi) within a specified amount of time and generated time histories of closed loop state and control responses. I have to plot its phase portrait using this code: And if first line was 9x - 21y - 10z + 5 where would 5 goes in the matrix? matlab. Stability Analysis for ODEs Marc R. The following plots have been produced with octave using the above procedure:. m — phase portrait plus graph of second order ordinary differential equation phasem. Phase portraits are an invaluable tool in studying dynamical systems. We have fixed at 3 and have observed the different behavior of the time evaluation and phase portraits as the parameter increases. Time is plot along the abscissa and x, y is plot along the ordinate. trajectories in the phase plane approach ~0 along curves that are tangent to the eigenvector of the eigenvalue closest to zero. Our second jaunt into the qualitative solution realm is the phase portrait. In the phase portrait below, every point on the green line is an equilibrium solution. sketching phase portraits. We will plot the derivatives as a vector at each (y1, y2) which will show us the initial direction from each point. a) b) c) 2 Consider the system defined by. 0 : Return to Main Page. Consider the homogeneous linear first-order system differential equations x'=ax+by y'=cx+dy. In practice, the width is obtained by using an image. Note that if, in your example, the roots were -1 and -2 (distinct but with same sign), the phase portrait would be composed of parabolas - confirm by yourself. Applications of Linear Algebra Basic Linear Systems and Matrices Cramer's Rule Determinant of a Matrix Dot Product Existence and Uniqueness of Solutions (Linear Equations) Finding the Inverse of a Square Matrix Gram-Schmidt Process Linear Equations Lines and Planes One-to-one Functions Onto Functions Row Reduction (Gaussian Elimination) Systems. Since in the nonlinear sytems the number of equilibrium points is not uniquie as in linear systems the. By plotting several trajectories you will get a preciser idea of phase diagram associated with. plot may behave differently depending on the environment in which it is invoked. 1BestCsharp blog 4,131,650 views. Parvan, no. points and locally (in a small region around the critical point), the phase portrait will resemble the corresponding picture. Inset: Time-lapse images of a passive particle rolling in a flow. Real part of is positive and the FP is unstable. 03SC (Alternatively, make the change of variables x 1 = x − x 0, y 1 = y − y 0, and drop all terms having order higher than one; then A is the matrix of coefﬁcients for the linear terms. However, there is one idea, not mentioned in the book, that is very useful to sketching and analyzing phase. Note that the right-hand expressions are not continuously differentiable at $(0,0)$ and there is no linearization at this point, which explains that while stable, the point $(0,0)$ does not look (after close examination) like a standard node. Also included are some examples of each kind so you can get an idea of what the homework is asking you to sketch for the phase portrait. Note that the used values of parameters are same as those used by Yoshisuke Ueda when he found chaos in 1961 (Ueda, 1979, 1980, and 1992). Try 3D plots, equations, inequalities, polar and parametric plots. For system (1), especially assuming that detA6= 0, this problem. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. Classify the xed points and determine their stability, when possible. Stability of linear and nonlinear systems. We will use Matplotlib's colormap to define colors for the trajectories. edu Michael T. Notice that this equation is autonomous! The phase portraits of these linear systems display a startling variety. Nonlinear Methods for the Social Sciences Stephen J. Then we will analyze the geometric effect of transforming a linear map to its Jordan Normal Form. But they are all limited in some ways. View PNG (It opens in a new tab, so you may need to allow popups). In chapter 2, we spent some time thinking about the phase portrait of the simple pendulum, and concluded with a challenge: can we design a nonlinear controller to reshape the phase portrait, with a very modest amount of actuation, so that the upright fixed point becomes globally stable? With unbounded torque, feedback linearization solutions (e. Introduction Infectious diseases have tremendous influence on human life. The system of equations is written u’ = AA*u, where AAis a given 2×2-matrix anduis a column vector. This is a linear equation, which we can solve completely. For math, science, nutrition, history. 01: 1; plot(x, 2 * min(x, 1 - x)) axis equal axis([0 1 0 1]) Note that this is not a phase plot in the standard sense, which is a plot of the system state over time, nor a phase portrait, which describes the structure of phase space (and which does not really apply to maps). 6 Asymptotic Behavior Sec. y 2 ' = y 1 + 4 y 2 + y 2 2. ams 20 ma tla no ucsc solving systems of first order equations with ode45 2015, yonatan katznelson the numerical solver, ode45 is designed to work with first. Hence, the two eigenvalues are opposite signs. If is a function, then the \emph{graph} of this function is the collection of points in the plane, where is the domain of. Konstantin Zuev (USC) Math 245, Lecture 9 January 30, 2012 5 / 15. 03, Spring, 1999 It is convenient to represent the solutions to an autonomous system ~x0= f~(~x)(where ~x= x y ) by means of a phase portrait. (Leave out the special case when one of the eigenvalues is 1. There have been some options of phase portraits already. In short, the best method for the analysis of a second order system is to plot a phase portrait. In this section we will solve systems of two linear differential equations in which the eigenvalues are real repeated (double in this case) numbers. Have a look at ShowColormaps if you want more information. In linear algebra, an ordered pair of numbers typically represents a vector, which in the present setting of coordinates can be taken to mean a change in location. Two areas where progress is being made are motion planning for mobile robots of factory floors (or on the surface of Mars), and control of highly articulated robots—such as. For part (c), use Mathematica to draw a phase portrait with numerous solutions for the system (3. System Upgrade on Feb 12th During this period, E-commerce and registration of new users may not be available for up to 12 hours. MOL 410/510: Introduction to Biological Dynamics Fall 2012 Problem Set #4, Nonlinear Dynamical Systems (due 10/19/2012) 6 MUST DO Questions, 1 OPTIONAL question 1. For a linear system, you can find the eigenvalues of matrix and the corresponding eigenvectors. Phase portraits in two dimensions 18. mass-spring-damper model in phase variable form. TFY4305 solutions extra exercises 2014 Phase portrait of problem 6. This video is unavailable. Again, the southern Barents–Kara metrics display similar behavior, which we choose not to plot for visual clarity. * Updated many function names to be more consistent with python coding. • Used MATLAB to sketch phase portraits and approximated changing behavior of dynamic systems in the neighborhood of the fixed point, then identified types of bifurcation • Nondimensionalized. Hannah Bardolaza, Maria Angela Faustino, Alexander De Los Reyes, Victor DC Vistro, Neil Irvin Cabello, Ivan Cedrick Verona, John Paul Ferrolino, Gerald Angelo Catindig, Karl Cedric Gonzales, Kerphy Liandro Patrocenio, Rommel Jagus, Elizabeth Ann Prieto, Deborah Anne Lumantas, John Daniel Vasquez, Jessica Afalla, Joselito Muldera, Valynn Katrine Mag-usara, Arnel Salvador, Armando Somintac. If the motion is outward, while if it is inward. MAPLETM LAB MANUAL FOR MATH 237 Diﬀerential Equations and Computer Methods Written by CESAR O. Since stable and unstable equilibria play quite different roles in the dynamics of a system, it is useful to be able to classify equi-librium points based on their stability. You can also plot the vector field associated with the system using quiver function. Example: Consider the harmonic oscillator equation. Phase portraits for 2 × 2 systems. 1 x x 1 2 v 2 v 1-1 1-1 Phase portraits for 2 × 2 systems. Left plot is for a= 1 and right plot is for a= 1. That is, it makes a few assumptions about the sampling rates that may not be evident to the average. A Javascript app to display the slope field for an ordinary differential equation, or the direction field (phase plane) for a two-variable system, and plot numerical solutions (e. The solution is on the ray in the opposite direction. 1 Linear equations Solving linear systems of equations is straightforward using the numpy submodule linalg. Sketch the phase portrait of this system. See examples/geneticswitch. • To plot 4(b), we need only be able to compute ˙x(x), it is not necessary to solve the differential equation. A phase portrait of a simple harmonic oscillator x 2 x 0. Note: Worksheets are in Maple 8 files and need the software to run. 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). A phase portrait is a plot of a segment’s angular displacement versus its angular velocity (Barela et al. We ﬂnd these. m — phase portrait plus graph of second order ordinary differential equation phasem. The procedure of dividing one equation by other other is quite general. We will be determining qualitative features of a dis-crete dynamical system of homogeneous di erence equations with constant coe cients.