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¶

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

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

Properties¶

Lyapunov spectrum

$+0.0411,\; +0.009651,\; -1.251$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.041$

Divergence ∇·f

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

Equilibria

5 equilibria
0 stable · 5 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Wang, Sun, van Wyk, Qi & van Wyk (2009), Braz. J. Phys. 39

BibTeX

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