Oregonator

Systems / ODEs / Chemical & biological systems

The Field–Körös–Noyes model of the Belousov–Zhabotinsky reaction — chemical relaxation oscillations.

continuous · ODE3 dimensionsrelaxation

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - x + y \\ \dot{y} &= \frac{q z - y z + y \left(1 - y\right)}{\epsilon} \\ \dot{z} &= \frac{f x - q z - y z}{\mu} \end{aligned} \]

Parameters

Symbol	Default	Role
q	0.0002	scaled rate-constant ratio of the bromide-consuming steps
f	1	stoichiometric factor for catalyst regeneration (bifurcation parameter)
mu	1e-06	timescale-separation parameter (stiffness)
epsilon	0.01	timescale-separation parameter (stiffness)

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = -1 - \frac{q}{\mu} - \frac{y}{\mu} - \frac{2 y}{\epsilon} - \frac{z}{\epsilon} + \frac{1}{\epsilon}$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Oregonator()
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

Field & Noyes (1974), J. Chem. Phys. 60, 1877-1884
doi:10.1063/1.1681288

BibTeX

@misc{oregonator,
  title = {Oregonator system},
  note = {Field & Noyes (1974), J. Chem. Phys. 60, 1877-1884},
  doi = {10.1063/1.1681288}
}