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

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

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

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.ForcedBrusselator()
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

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}
}