Phase portrait of van der pol equation pdf

In this paper we study the global phase portrait of the normal form of a degenerate bogdanovtakens system with symmetry, i. For a single dof mechanical system, the phase plane is in fact is. This equation exhibits a simple stable attracting critical point and two unstable equilibria. The time in the equation has been scaled so that the frequency associated with the spring force alone is unity. The companion system is x y y x a x21 y this system turns out to continue to have periodic solutions. 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. Generic nonlinear oscillator model differential equations of this kind can be studied by existing analytical methods 3, 4. Phase portrait of lotkavolterra equation wusu ashiribo senapon and akanbi moses adebowale. This can be inferred geometrically from its associated phase portraits. You need to calculate the vector field at every point you want an arrow to be shown. Before anything can be done with a differential equation, a socalled. The time in the equation has been scaled so that the frequency associated. In particular, equation 1 serves after making several simplifying assumptions as a mathematical model of a generator on a triode for a tube with a cubic characteristic. We begin our numerical work with a phase portrait based on four selected.

The equation models a nonconservative system in which energy is added to and subtracted from the system. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. One might think of this as a model for a springmass system where the damping force ux depends on position for example, the mass might be moving through a viscous medium. Introduction to singular perturbation methods nonlinear. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Click the phase plane plot to set the initial conditions for and. The first system is a linear system, so its phase portrait can be fully determined from the eigenvector structure of the associated matrix. Solutions to assignment 5 department of mathematics. The phase plane form of this equation can be written as x 12x 2 x 212 121, 1 fx x x x x. The control objective is to make the origin globally asymptotically stable. Each set of initial conditions is represented by a different curve, or point.

A copy of the license is included in the section entitled gnu free documentation license. The differential equation of the orthogonal curves is solved here exactly in terms of modified bessel functions. It may be best to think of the system of equations as the single vector equation x y fx,y gx,y. The classical experimental setup of the system is the oscillator with vacuum triode. Phase portrait and field directions of twodimensional linear systems of odes santos bravo yuste. A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Use pplane to draw the phase portraits of the damped anharmonic oscillator. The above equation will be solved numerically using matlabs ode45 for di. It describes many physical systems collectively called vanderpoloscillators. After a relatively brief time, the shapes of the curves become very close to those shown in figure 5. For a stiff problem, solutions can change on a time scale that is very short compared to the interval of integration, but the solution of interest changes on a much longer time scale. Phase line for an autonomous equation for a single autonomous equation y0 fy we used a phase line to illustrate the equilibrium solutions.

Phase space of an intermittently driven oscillator manu p. Using matlab to solve differential equations numerically. Derive equation 2 by applying newtons second law of motion to the bob the pendulum. The limit cycle begins as circle and, with varying. The array has 3033 rows and 1 column so this is the number of steps. Lecture notes on nonlinear vibrations mathematics faculty. A nonlinear second order ode was solved numerically using matlabs ode45. First, we assume that is known and design stabilizing control laws by the feedback linearization method 2. The ideas in two dimensions are the same, but the behavior can be far more complicated. The same is true for the orthogonal trajectories of this family.

We discuss the use of phase plane plots for understanding the qualitative behavior of systems of differential equations. The solution of this system of two differential equations gives the trajectory in the phase space, that is, the x1, x2 plane. The slope of a trajectory in the phase plane is 2 22 12 1 11 2 dx x x x x1 dx x x. Before proceeding, we recommend that you test out the introductory example ch1riccati.