Bouali¶

Systems / ODEs / Coupled systems

The Bouali business-cycle flow (first regime) — a feedback-augmented Van der Pol oscillator tracing a stretched-loop economic 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`	`1`	oscillator self-excitation
`b`	`-0.3`	z→x feedback gain
`bb`	`1`	x·z cross-feedback
`c`	`0.05`	z linear decay
`g`	`1`	nonlinear damping
`m`	`1`	z-loop drive
`y0`	`4`	reference level of x

Properties¶

Lyapunov spectrum

$+0.01099,\; +0.02233,\; -0.1929$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.173$

Divergence ∇·f

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

Equilibria

4 equilibria
0 stable · 4 unstable

Define it in TSDynamics¶

import tsdynamics as ts

sys = ts.systems.Bouali()
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{bouali,
  title = {Bouali system},
  note = {Bouali (1999), Int. J. Bifurcation Chaos 9, 745-756}
}