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

\[ \begin{aligned} \dot{y_{0}} &= - a y_{0} + a y_{1} \\ \dot{y_{1}} &= c y_{1} - y_{0} y_{2} + y_{0} \left(- a + c\right) \\ \dot{y_{2}} &= - b y_{2} + y_{0} y_{1} + y_{0} y_{2} - y_{1} y_{2} - y_{3} \\ \dot{y_{3}} &= - d y_{3} - y_{0} y_{2} + y_{1} y_{2} \end{aligned} \]

Parameters

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

Properties

Lyapunov spectrum

$+1.223,\; +0.01363,\; -15.28,\; -37.95$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.081$

Divergence ∇·f

$\nabla\!\cdot f = - a - b + c - d + y_{0} - y_{1}$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Meier (2003), Presentation of Attractors with Cinema

BibTeX

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