The Forced van der Pol Equation

Warren Weckesser


One form of the periodically forced van der Pol equation is

We are interested in the case where μ is large.

By making the definitions

    
    
and

we can convert the second order differential equation into the following autonomous three dimensional system:



This is the system studied by me and my collaborators John Guckenheimer and Kathleen Hoffman. Our results are reported in publications 4, 6, 8, and 9 listed in my CV.

The following plot shows an example of the phase space of this system.

The yellow curve is a solution to the system of equations when ε is small but positive. The blue surface is the critical manifold and the white vector field shows the slow subsystem that results when ε = 0. The red and green markers are folded equilibria. These are equilibria of the "desingularized" slow subsystem, but they are not true equilibria of the original equations when ε > 0. The red markers are folded saddles, and the green markers are folded spiral points.

In publication 8 (see also 9), we studied a Poincare map defined as in the following plot:

Plot showing Poincare map
We computed the map from Σ1 to Σ2 by solving the differential equations. The map was completed by using a symmetry of the differential equations to map Σ2 to Σ1.

The following plot shows a horseshoe map that occurs in this Poincare map. The images of the short segments at the top and bottom of the red parallelogram are plotted in cyan. (There are two cyan curves in the plot, but they are very close together.)

Plot
This plot was computed using AUTO, by setting up the computation of the Poincare map as a boundary value problem, and by doing continuation in the initial conditions.