Van der Pol Oscillator

anypinn create my-project --template van-der-pol

Second-order nonlinear oscillator. Recovers nonlinearity parameter \(\mu\).

Background

The Van der Pol oscillator, proposed by Balthasar van der Pol (1926) to model oscillations in vacuum tube circuits, is a canonical example of a self-excited oscillator. When \(\mu > 0\) the system has negative damping for small amplitudes (energy injection) and positive damping for large amplitudes (energy dissipation), producing a stable limit cycle, a self-sustaining periodic orbit whose shape and amplitude are independent of initial conditions. For small \(\mu\) the waveform is nearly sinusoidal; for large \(\mu\) it becomes a relaxation oscillation with alternating slow drifts and rapid jumps.

Governing Equations

\[ \ddot{x} - \mu\,(1 - x^2)\,\dot{x} + x = 0 \]

Equivalently, with \(v = \dot{x}\):

\[ \begin{cases} \dfrac{dx}{dt} = v \\[8pt] \dfrac{dv}{dt} = \mu\,(1 - x^2)\,v - x \end{cases} \]

where:

• \(x(t)\): displacement (state variable)
• \(v(t)\): velocity
• \(\mu\): nonlinearity / damping strength (to recover)

Default Configuration

The generated template uses the following values.

Parameters to recover:

Symbol	Code constant	True value
\(\mu\)	TRUE_MU	\(1.0\)

Initial conditions: \(x(0) = 2.0, \quad \dot{x}(0) = 0.0\)

Domain: \(t \in [0, 20]\) s

Features Demonstrated

• Second-order ODE support via ODEProperties.order
• Nonlinear dynamics with limit cycles
• Scalar Parameter recovery

Results