GenesioTesi

Systems / ODEs / Chaotic attractors

A canonical jerk control system whose single quadratic nonlinearity, analysed by harmonic balance, produces a Shilnikov strange attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= y \\ \dot{y} &= z \\ \dot{z} &= - a z - b y - c x + x^{2} \end{aligned} \]

Parameters

Symbol	Default	Role
a	0.44	acceleration feedback gain
b	1.1	velocity feedback gain
c	1	position feedback gain

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a$
constant

Equilibria

2 equilibria
0 stable · 2 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.GenesioTesi()
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

Genesio & Tesi (1992), Automatica 28, 531-548
doi:10.1016/0005-1098(92)90177-h

BibTeX

@misc{genesiotesi,
  title = {GenesioTesi system},
  note = {Genesio & Tesi (1992), Automatica 28, 531-548},
  doi = {10.1016/0005-1098(92)90177-h}
}