Selkov

Systems / ODEs / Chemical & biological systems

The minimal glycolytic clock — one product-activation loop, and a limit cycle for every substrate supply inside the Hopf window.

continuous · ODE2 dimensionslimit cycle

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Definition

\[ \begin{aligned} \dot{x} &= a y + x^{2} y - x \\ \dot{y} &= - a y + b - x^{2} y \end{aligned} \]

Parameters

Symbol	Default	Role
a	0.1	scaled reverse rate of the enzyme (oscillations need a < 1/8)
b	0.6	scaled substrate supply rate — the bifurcation parameter

State variables: x, y

Properties

Lyapunov spectrum

≥ 0 positive exponents (literature — analytic: Poincare-Bendixson forbids chaos in a planar autonomous flow)

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a - x^{2} + 2 x y - 1$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Selkov()
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

Sel'kov (1968), Eur. J. Biochem. 4, 79-86
doi:10.1111/j.1432-1033.1968.tb00175.x

BibTeX

@misc{selkov,
  title = {Selkov system},
  note = {Sel'kov (1968), Eur. J. Biochem. 4, 79-86},
  doi = {10.1111/j.1432-1033.1968.tb00175.x}
}