Non Linear Limit Cycles, Van Der Pol Equation.

Cuthbert Nyack

The Applet below shows the sinusoidal response of the Nonlinear Van Der Pol eqn and illustrates some aspects of the behaviour of nonlinear systems.

Harmonic and subharmonic responses.

As the applied frequency is changed the response goes through several subharmonic and harmonic "resonances" a few are shown below.
eg parameters (1.0, 0.6, 1.0, w, 10.0, 1.0)
w = 0.35, ratio = 5:2
w = 0.43, ratio = 2:1
w = 0.56, ratio = 3:2
w = 1.24, ratio = 4:5
w = 1.3, ratio = 3:4
w = 1.44, ratio = 2:3
w = 1.58, ratio = 3:5
w = 1.89, ratio = 1:2
w = 2.35, ratio = 2:5

Entrainment.

For the same eg parameters, the response frequency is entrained by the applied frequency for w from 0.72 to 1.14.

The Applet below shows the pulse response of the Nonlinear Van Der Pol eqn. and illustrates one aspect of the limit cycle.

The height of the initial applied pulse affects the time for the limit cycle to be established but does not affect the amplitude of the cycle.

The Applet below shows the response of the Nonlinear Van Der Pol eqn. to varying Initial Conditions and illustrates another aspect of the limit cycle.

Varying I.C. shows that the limit cycle is independent of whether the system starts outside or inside the limit cycle in the phase plane(shown in yellow).

Gif image below shows how applet should appear.

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