Van Der Pol

The van der Pol Oscillator

In the 1920's dutch physicist Balthasar van der Pol experimented with electrical circuits containing vacuum tubes. Despite there being no direct voltage forcing of the circuit, he noticed stable oscillations occurring. An equation that models the behavior van der Pol circuit is:

${d^2 V}/{dt^2} - {1}/{C}(α-3γV^2){dV}/{dt} + {1}/{LC}V = 0$

By scaling time and voltage, this can be reduced to the following cannonical form:

${d^2V}/{dt^2}-μ(1-V^2){dV}/{dt}+V=0$

To convert this to first order form, a current variable $I={dV}/{dt}$ is defined. The system of ordinary differential equations (ODEs) is then written as:

${dV}/{dt} = I$
${dI}/{dt} = μ(1-V^2)·I - V$

In vector field form this is represented as:

${W(V,I) = {dV}/{dt} {∂}/{∂V} + {dI}/{dt} {∂}/{∂I} = I {∂}/{∂V} + (μ(1-V^2)·I - V) {∂}/{∂I}}$

When the solution is plotted relative to axes V and I, the stable oscillations appear as a closed loop. This solution is referred to as a limit cycle because solutions that start inside or outside the loop will approach the closed loop limit cycle solution as time goes to infinity.

For the van der Pol plot below, a μ value of 0.2 is used. Model contours are shown for a function that satisifes the 1-D modal equation:

$W(f) = {dV}/{dt} {∂f}/{∂V} + {dI}/{dt} {∂f}/{∂I} = I {∂f}/{∂V} + (0.2(1-V^2)I - V) {∂f}/{∂I} = 1$

The 1-D modal function $f$ can be calculated numerically by taking the limit cycle solution and expanding the solution curve slightly in the I, V space (i.e. scaling each point by 1.01). Solutions for each point on the expanded curve are then solved backwards in time (i.e. by taking negative time steps in the ODE solver). Similarly, to calculate modal contours inside the limit cycle, the limit cycle was contracted a few percent and the solution was calculated backwards for each point on the contracted limit cycle.

The limit cycle solution is plotted in black. The 'Show Time Evolution' button starts a forward time animation of the modal contours, or a sample of the ODE solutions used to calculate the modal contours, or both. Modal contours outside the limit cycle are drawn at equal timesteps $t=i·1.8$ where i has integer values from 0 to 9, while contours on the inside of the limit cycle drawn at equal timestep that are slightly larger 1.8.

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