GuckenheimerHolmes

Systems / ODEs / Chaotic attractors

A symmetric flow with a structurally stable heteroclinic cycle — the prototype of cycling chaos, slowing near each saddle in turn.

continuous · ODE3 dimensionscycling chaos

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= a x - b y + c x z + d x^{2} z + d y^{2} z \\ \dot{y} &= a y + b x + c y z \\ \dot{z} &= - a z^{3} + e - f x^{2} - f y^{2} - z^{2} \end{aligned} \]

Parameters

Symbol	Default	Role
a	0.4	linear growth/rotation coefficient
b	20.25	linear growth/rotation coefficient
c	3	linear growth/rotation coefficient
d	1.6	nonlinear coupling coefficient
e	1.7	nonlinear coupling coefficient
f	0.44	nonlinear coupling coefficient

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - 3 a z^{2} + 2 a + 2 c z + 2 d x z - 2 z$
state-dependent

Equilibria

2 equilibria
0 stable · 2 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.GuckenheimerHolmes()
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

Guckenheimer & Holmes (1988), Math. Proc. Camb. Phil. Soc. 103, 189-192
doi:10.1017/s0305004100064732

BibTeX

@misc{guckenheimerholmes,
  title = {GuckenheimerHolmes system},
  note = {Guckenheimer & Holmes (1988), Math. Proc. Camb. Phil. Soc. 103, 189-192},
  doi = {10.1017/s0305004100064732}
}