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

\[ \begin{aligned} \dot{y_{0}} &= 3 \gamma \mu y_{0} - \gamma y_{0}^{3} + \gamma y_{1} \\ \dot{y_{1}} &= \beta - 2 \mu y_{0} - y_{1} - y_{2} \\ \dot{y_{2}} &= \kappa y_{0} - \kappa y_{2} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.01412,\; +0.000976,\; -0.3106$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.049$

Divergence ∇·f

$\nabla\!\cdot f = 3 \gamma \mu - 3 \gamma y_{0}^{2} - \kappa - 1$
state-dependent

Equilibria

2 equilibria
0 stable · 2 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Strizhak & Kawczynski (1995), J. Phys. Chem. 99, 10830-10833

BibTeX

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