WangSun

Systems / ODEs / Coupled systems

A Wang–Sun quadratic flow carrying one cross-product nonlinearity per equation (y*z, x*z, x*y), producing a chaotic attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	0.2	self-feedback gain on the x channel
b	-0.01	linear x-into-y coupling gain
d	-0.4	self-feedback gain on the y channel
e	-1	self-feedback gain on the z channel
f	-1	x*y cross-product strength in the z equation
q	1	y*z cross-product strength in the x equation

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = a + d + e$
constant

Equilibria

5 equilibria
0 stable · 5 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.WangSun()
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

Wang, Sun, van Wyk, Qi & van Wyk (2009), Braz. J. Phys. 39
doi:10.1590/s0103-97332009000500007

BibTeX

@misc{wangsun,
  title = {WangSun system},
  note = {Wang, Sun, van Wyk, Qi & van Wyk (2009), Braz. J. Phys. 39},
  doi = {10.1590/s0103-97332009000500007}
}