HyperRossler

Systems / ODEs / Chaotic attractors

Rössler's original four-dimensional flow — the first system shown to be hyperchaotic, with two positive Lyapunov exponents from a single quadratic term.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, z, w)

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	0.25	flow coefficient
b	3	flow coefficient
c	0.5	flow coefficient
d	0.05	flow coefficient

Properties

Lyapunov spectrum

$+0.1392,\; +0.01223,\; +0.0101,\; -23.85$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3.007$

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Rössler (1979), Phys. Lett. A 71, 155-157

BibTeX

@misc{hyperrossler,
  title = {HyperRossler system},
  note = {Rössler (1979), Phys. Lett. A 71, 155-157}
}