Phase portrait of van der pol equation pdf

Phase space of an intermittently driven oscillator manu p. The above equation will be solved numerically using matlabs ode45 for di. Use pplane to draw the phase portraits of the damped anharmonic oscillator. After a relatively brief time, the shapes of the curves become very close to those shown in figure 5. 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. The companion system is x y y x a x21 y this system turns out to continue to have periodic solutions. 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.

A copy of the license is included in the section entitled gnu free documentation license. The limit cycle begins as circle and, with varying. A nonlinear second order ode was solved numerically using matlabs ode45. 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. This equation exhibits a simple stable attracting critical point and two unstable equilibria. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1.

The equation models a nonconservative system in which energy is added to and subtracted from the system. Phase portrait of lotkavolterra equation wusu ashiribo senapon and akanbi moses adebowale. This can be inferred geometrically from its associated phase portraits. We discuss the use of phase plane plots for understanding the qualitative behavior of systems of differential equations. It may be best to think of the system of equations as the single vector equation x y fx,y gx,y. The array has 3033 rows and 1 column so this is the number of steps. Each set of initial conditions is represented by a different curve, or point. Before anything can be done with a differential equation, a socalled. The time in the equation has been scaled so that the frequency associated. Before proceeding, we recommend that you test out the introductory example ch1riccati. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The slope of a trajectory in the phase plane is 2 22 12 1 11 2 dx x x x x1 dx x x. Lecture notes on nonlinear vibrations mathematics faculty.

In this paper we study the global phase portrait of the normal form of a degenerate bogdanovtakens system with symmetry, i. The time in the equation has been scaled so that the frequency associated with the spring force alone is unity. Click the phase plane plot to set the initial conditions for and. The same is true for the orthogonal trajectories of this family. Phase portrait and field directions of twodimensional linear systems of odes santos bravo yuste. The solution of this system of two differential equations gives the trajectory in the phase space, that is, the x1, x2 plane.

We begin our numerical work with a phase portrait based on four selected. The classical experimental setup of the system is the oscillator with vacuum triode. Generic nonlinear oscillator model differential equations of this kind can be studied by existing analytical methods 3, 4. You need to calculate the vector field at every point you want an arrow to be shown. Derive equation 2 by applying newtons second law of motion to the bob the pendulum. Using matlab to solve differential equations numerically. It describes many physical systems collectively called vanderpoloscillators. The first system is a linear system, so its phase portrait can be fully determined from the eigenvector structure of the associated matrix. Phase line for an autonomous equation for a single autonomous equation y0 fy we used a phase line to illustrate the equilibrium solutions. Solutions to assignment 5 department of mathematics.