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¶

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

Parameters¶

Symbol	Default	Role
`a`	`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

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

4 equilibria
0 stable · 4 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Bouali()
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{bouali,
  title = {Bouali system},
  note = {Bouali (1999), Int. J. Bifurcation Chaos 9, 745-756},
  doi = {10.1142/s0218127499000535}
}