AnishchenkoAstakhov

Systems / ODEs / Oscillatory systems

A radiophysical self-oscillator with inertial nonlinearity — the canonical model of spiral (saddle-focus) chaos reached by the Feigenbaum period-doubling route.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= \mu x - x z + y \\ \dot{y} &= - x \\ \dot{z} &= \frac{\eta x^{2} \left(\operatorname{sign}{\left(x \right)} + 1\right)}{2} - \eta z \end{aligned} \]

Parameters

Symbol	Default	Role
eta	0.5	inertial-relaxation rate of the nonlinear element
mu	1.2	excitation parameter (negative damping of the circuit)

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - \eta + \mu - z$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.AnishchenkoAstakhov()
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

Anishchenko et al. (2007), Nonlinear Dynamics of Chaotic and Stochastic Systems
doi:10.1007/978-3-540-38168-6

BibTeX

@misc{anishchenkoastakhov,
  title = {AnishchenkoAstakhov system},
  note = {Anishchenko et al. (2007), Nonlinear Dynamics of Chaotic and Stochastic Systems},
  doi = {10.1007/978-3-540-38168-6}
}