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

\[ \begin{aligned} \dot{y_{0}} &= a y_{0} y_{0} - a y_{0} y_{1} - b y_{2} \\ \dot{y_{1}} &= g y_{0}^{2} y_{1} - g y_{1} \\ \dot{y_{2}} &= bb m y_{0} y_{2} - c y_{2} - 1.5 m y_{0} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.003031,\; +0.01853,\; -0.01317$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3$

Divergence ∇·f

$\nabla\!\cdot f = a y_{0} - a y_{1} + bb m y_{0} - c + g y_{0}^{2} - g$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Bouali2()
traj = sys.integrate(final_time=100.0, dt=0.01)

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

Reference

Bouali (1999), Int. J. Bifurcation Chaos 9, 745-756

BibTeX

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