KawczynskiStrizhak

Systems / ODEs / Population dynamics

A three-variable polynomial reduction of the Belousov-Zhabotinsky reaction reproducing its mixed-mode and period-doubling routes to chaos.

continuous · ODE3 dimensionschaotic, bursting

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= 3 \gamma \mu x - \gamma x^{3} + \gamma y \\ \dot{y} &= \beta - 2 \mu x - y - z \\ \dot{z} &= \kappa x - \kappa z \end{aligned} \]

Parameters

Symbol	Default	Role
beta	-0.4	constant drive (offset) of the slow recovery variable
gamma	0.49	timescale / gain of the fast cubic nullcline
kappa	0.2	relaxation rate of the slowest variable
mu	2.1	bifurcation parameter shaping the cubic nullcline

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 3 \gamma \mu - 3 \gamma x^{2} - \kappa - 1$
state-dependent

Equilibria

2 equilibria
0 stable · 2 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.KawczynskiStrizhak()
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

Strizhak & Kawczynski (1995), J. Phys. Chem. 99, 10830-10833
doi:10.1021/j100027a024

BibTeX

@misc{kawczynskistrizhak,
  title = {KawczynskiStrizhak system},
  note = {Strizhak & Kawczynski (1995), J. Phys. Chem. 99, 10830-10833},
  doi = {10.1021/j100027a024}
}