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

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

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

State variables: alpha, beta, gamma

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = \frac{2 \alpha \beta}{\sigma} - \beta^{2} - 1 - \frac{1}{\sigma} - \frac{1}{\delta}$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.IsothermalChemical()
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

Petrov, Scott & Showalter (1992), J. Chem. Phys. 97, 6191-6198
doi:10.1063/1.463727

BibTeX

@misc{isothermalchemical,
  title = {IsothermalChemical system},
  note = {Petrov, Scott & Showalter (1992), J. Chem. Phys. 97, 6191-6198},
  doi = {10.1063/1.463727}
}