Oregonator

Systems / ODEs / Chemical & biological systems

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

continuous · ODE3 dimensionsrelaxation

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Definition

\[ \begin{aligned} \dot{y_{0}} &= - y_{0} + y_{1} \\ \dot{y_{1}} &= \frac{q y_{2} - y_{1} y_{2} + y_{1} \left(1 - y_{1}\right)}{\epsilon} \\ \dot{y_{2}} &= \frac{f y_{0} - q y_{2} - y_{1} y_{2}}{\mu} \end{aligned} \]

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)

Properties

Lyapunov spectrum

$+0.0208,\; -229.6,\; -359.1$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Field & Noyes (1974), J. Chem. Phys. 60, 1877-1884

BibTeX

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