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

\[ \begin{aligned} \dot{x} &= - y - z \\ \dot{y} &= a y + w + x \\ \dot{z} &= b + x z \\ \dot{w} &= - c z + d w \end{aligned} \]

Parameters

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

State variables: x, y, z, w

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = a + d + x$
state-dependent

Equilibria

2 equilibria
0 stable · 2 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HyperRossler()
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

Rössler (1979), Phys. Lett. A 71, 155-157
doi:10.1016/0375-9601(79)90150-6

BibTeX

@misc{hyperrossler,
  title = {HyperRossler system},
  note = {Rössler (1979), Phys. Lett. A 71, 155-157},
  doi = {10.1016/0375-9601(79)90150-6}
}