ForcedBrusselator

Systems / ODEs / Chemical & biological systems

The Brusselator under periodic forcing — entrainment and chaos in an autocatalytic reaction.

continuous · ODE3 dimensionsforced

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	0.4	constant feed of the activator x
b	1.2	control parameter (Hopf at b = 1 + a^2)
f	0.05	amplitude of the periodic forcing
w	0.81	angular frequency of the forcing

Properties

Lyapunov spectrum

$+0.04712,\; -0.244,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.193$

Divergence ∇·f

$\nabla\!\cdot f = - b - y_{0}^{2} + 2 y_{0} y_{1} - 1$
state-dependent

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Prigogine (1980), From Being to Becoming, W.H. Freeman

BibTeX

@misc{forcedbrusselator,
  title = {ForcedBrusselator system},
  note = {Prigogine (1980), From Being to Becoming, W.H. Freeman}
}