HyperCai¶

Systems / ODEs / Exotic systems

A 4-D hyperchaotic flow augmenting a Lorenz-like chaotic core with a fourth feedback state, yielding two positive Lyapunov exponents.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, y, w)

Interactive: drag to rotate

Definition¶

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

Parameters¶

Symbol	Default	Role
`a`	`27.5`	linear coupling rate of the x–y subsystem
`b`	`3`	linear gain in the y equation
`c`	`19.3`	linear gain in the y equation
`d`	`2.9`	damping of the z mode (fed by y²)
`e`	`3.3`	feedback gain of the auxiliary state w onto x

Properties¶

Lyapunov spectrum

$+1.624,\; +0.1088,\; -0.004117,\; -12.83$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3.135$

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Cai & Huang (2007), Int. J. Nonlinear Sci.

BibTeX

@misc{hypercai,
  title = {HyperCai system},
  note = {Cai & Huang (2007), Int. J. Nonlinear Sci.}
}