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

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Definition

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

Parameters

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

Properties

Lyapunov spectrum

$+0.03327,\; -0.007017,\; -0.4509$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.058$

Divergence ∇·f

$\nabla\!\cdot f = - \eta + \mu - y_{2}$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Anishchenko et al. (2007), Nonlinear Dynamics of Chaotic and Stochastic Systems

BibTeX

@misc{anishchenkoastakhov,
  title = {AnishchenkoAstakhov system},
  note = {Anishchenko et al. (2007), Nonlinear Dynamics of Chaotic and Stochastic Systems}
}