VanDerPol

Systems / ODEs / Chemical & biological systems

The archetypal relaxation oscillator — nonlinear damping pumps small swings and bleeds large ones, so every orbit lands on one and the same limit cycle.

continuous · ODE2 dimensionslimit cycle

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= y \\ \dot{y} &= \mu y \left(1 - x^{2}\right) - x \end{aligned} \]

Parameters

Symbol	Default	Role
mu	1	nonlinear damping strength (0 = harmonic, >> 1 = relaxation oscillator)

State variables: x, y

Properties

Lyapunov spectrum

≥ 0 positive exponents (literature — analytic: Poincare-Bendixson forbids chaos in a planar autonomous flow)

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - \mu x^{2} + \mu$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.VanDerPol()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

van der Pol (1926), London Edinburgh Dublin Philos. Mag. J. Sci. 2, 978-992
doi:10.1080/14786442608564127

BibTeX

@misc{vanderpol,
  title = {VanDerPol system},
  note = {van der Pol (1926), London Edinburgh Dublin Philos. Mag. J. Sci. 2, 978-992},
  doi = {10.1080/14786442608564127}
}