HyperPang¶

Systems / ODEs / Exotic systems

A 4-D hyperchaotic flow built by adding a linear controller to the Lü system, with two positive Lyapunov exponents and a studied Hopf bifurcation.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, y, w)

Interactive: drag to rotate

Definition¶

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

Parameters¶

Symbol	Default	Role
`a`	`36`	coupling rate of the x–y (Lü) subsystem
`b`	`3`	damping of the z mode
`c`	`20`	self-gain of the y mode
`d`	`2`	feedback gain of x and y onto the auxiliary state w

Properties¶

Lyapunov spectrum

$+1.462,\; +0.05796,\; -0.004547,\; -20.52$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3.074$

Divergence ∇·f

$\nabla\!\cdot f = - a - b + c$
constant

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Pang & Liu (2011), J. Comput. Appl. Math. 235, 2775

BibTeX

@misc{hyperpang,
  title = {HyperPang system},
  note = {Pang & Liu (2011), J. Comput. Appl. Math. 235, 2775}
}