IsothermalChemical

Systems / ODEs / Chemical & biological systems

The Petrov–Scott–Showalter isothermal autocatalator — cubic autocatalysis feeding a slow species, the canonical route to mixed-mode oscillations and chaos in unforced chemistry.

continuous · ODE3 dimensionsmixed-mode chaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= \mu \left(\kappa + y_{2}\right) - y_{0} y_{1}^{2} - y_{0} \\ \dot{y_{1}} &= \frac{y_{0} y_{1}^{2} + y_{0} - y_{1}}{\sigma} \\ \dot{y_{2}} &= \frac{y_{1} - y_{2}}{\delta} \end{aligned} \]

Parameters

Symbol	Default	Role
delta	1	timescale ratio of the slow species
kappa	2.5	feedback coupling of the slow species
mu	0.29786	scaled inflow / precursor-decay rate
sigma	0.013	timescale ratio of the fast autocatalyst

Properties

Lyapunov spectrum

$-0.02424,\; +0.001694,\; -6.767$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1.07$

Divergence ∇·f

$\nabla\!\cdot f = - y_{1}^{2} - 1 + \frac{2 y_{0} y_{1}}{\sigma} - \frac{1}{\sigma} - \frac{1}{\delta}$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Petrov, Scott & Showalter (1992), J. Chem. Phys. 97, 6191-6198

BibTeX

@misc{isothermalchemical,
  title = {IsothermalChemical system},
  note = {Petrov, Scott & Showalter (1992), J. Chem. Phys. 97, 6191-6198}
}