ForcedVanDerPol

Systems / ODEs / Chemical & biological systems

Van der Pol under periodic forcing — frequency entrainment and the route to chaos.

continuous · ODE3 dimensionsforced

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	1.2	amplitude of the periodic forcing
mu	8.53	nonlinear damping strength
w	0.63	angular frequency of the forcing

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.ForcedVanDerPol()
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{forcedvanderpol,
  title = {ForcedVanDerPol system},
  note = {van der Pol (1926), London Edinburgh Dublin Philos. Mag. J. Sci. 2, 978-992},
  doi = {10.1080/14786442608564127}
}