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

\[ \begin{aligned} \dot{y_{0}} &= a y_{0} - b y_{1} + c y_{0} y_{2} + d y_{0}^{2} y_{2} + d y_{1}^{2} y_{2} \\ \dot{y_{1}} &= a y_{1} + b y_{0} + c y_{1} y_{2} \\ \dot{y_{2}} &= - a y_{2}^{3} + e - f y_{0}^{2} - f y_{1}^{2} - y_{2}^{2} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.4398,\; +0.008072,\; -0.8395$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.533$

Divergence ∇·f

$\nabla\!\cdot f = - 3 a y_{2}^{2} + 2 a + 2 c y_{2} + 2 d y_{0} y_{2} - 2 y_{2}$
state-dependent

Equilibria

2 equilibria
0 stable · 2 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Guckenheimer & Holmes (1988), Math. Proc. Camb. Phil. Soc. 103, 189-192

BibTeX

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