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

\[ \begin{aligned} \dot{y_{0}} &= y_{1} \\ \dot{y_{1}} &= a \sin{\left(y_{2} \right)} + \mu y_{1} \left(1 - y_{0}^{2}\right) - y_{0} \\ \dot{y_{2}} &= w \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.01472,\; -13.18,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.001$

Divergence ∇·f

$\nabla\!\cdot f = - \mu y_{0}^{2} + \mu$
state-dependent

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.ForcedVanDerPol()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

van der Pol (1926), London Edinburgh Dublin Philos. Mag. J. Sci. 2, 978-992

BibTeX

@misc{forcedvanderpol,
  title = {ForcedVanDerPol system},
  note = {van der Pol (1926), London Edinburgh Dublin Philos. Mag. J. Sci. 2, 978-992}
}