Bouali2

Systems / ODEs / Coupled systems

An extended Van der Pol oscillator with a chaos-inducing feedback loop, proposed by Bouali as an idealised macroeconomic business-cycle model with a stretched-loop strange attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - a x y + a x y_{0} - b z \\ \dot{y} &= g x^{2} y - g y \\ \dot{z} &= bb m x z - c z - 1.5 m x \end{aligned} \]

Parameters

Symbol	Default	Role
a	3	self-excitation gain of the oscillator core
b	2.2	feedback gain of the z channel into x
bb	0	z-loop x*z cross-product gain
c	0	z-loop linear decay gain
g	1	nonlinear damping of the y channel
m	-0.0026667	z-loop drive gain
y0	1	reference level of the x channel

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a y + a y_{0} + bb m x - c + g x^{2} - g$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Bouali2()
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

Bouali (1999), Int. J. Bifurcation Chaos 9, 745-756
doi:10.1142/s0218127499000535

BibTeX

@misc{bouali2,
  title = {Bouali2 system},
  note = {Bouali (1999), Int. J. Bifurcation Chaos 9, 745-756},
  doi = {10.1142/s0218127499000535}
}