HyperYan

Systems / ODEs / Chaotic attractors

A four-dimensional flow with multiple cross-product nonlinearities and a feedback variable producing hyperchaos.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (y, z, w)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - a x + a y \\ \dot{y} &= c y - x z + x \left(- a + c\right) \\ \dot{z} &= - b z - w + x y + x z - y z \\ \dot{w} &= - d w - x z + y z \end{aligned} \]

Parameters

Symbol	Default	Role
a	37	flow coefficient
b	3	flow coefficient
c	26	flow coefficient
d	38	flow coefficient

State variables: x, y, z, w

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a - b + c - d + x - y$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HyperYan()
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

Meier (2003), Presentation of Attractors with Cinema

BibTeX

@misc{hyperyan,
  title = {HyperYan system},
  note = {Meier (2003), Presentation of Attractors with Cinema}
}