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Van der Pol Oscillator

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This example shows how to model the second-order Van der Pol (VDP) differential equation in Simulink®. In dynamics, the VDP oscillator is non-conservative and has nonlinear damping. At high amplitudes, the oscillator dissipates energy. At low amplitudes, the oscillator generates energy. The oscillator is given by this second-order differential equation:

$$\frac{d^2 x}{dt^2} - Mu \left( 1- x^2 \right) \frac{dx}{dt} + x = 0$$

where:

  • x is position as a function of time.

  • Mu is damping.

The VDP oscillator is used in physical and biological sciences, including electric circuits.

open_system('vdp');

Simulate with Mu = 1

When Mu = 1, the VDP oscillator has nonlinear damping.

set_param('vdp/Mu','Gain','1')
sim('vdp');
open_system('vdp/Scope');

Simulate with Mu = 0

When Mu = 0, the VDP oscillator has no damping. Energy is conserved in this simple harmonic oscillator. The equation becomes:

$$\frac{d^2 x}{dt^2} + x = 0$$

set_param('vdp/Mu','Gain','0')
sim('vdp');
open_system('vdp/Scope');

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References

[1] Cartwbight, M. L. "Balthazar Van Der Pol." Journal of the London Mathematical Society. Wiley. s1 35 (July 1960): 367–376. https://doi:10.1112/jlms/s1-35.3.367.

[2] Hirsch, Morris W., Stephen Smale, Robert L. Devaney, and Morris W. Hirsch. Differential Equations, Dynamical Systems, and an Introduction to Chaos. 2nd Ed. San Diego: Academic Press, 2004.