Rossler

Systems / ODEs / Chaotic attractors

A single-folded-band attractor built from the minimal nonlinearity — one quadratic term — yet fully chaotic.

continuous · ODE3 dimensionschaoticdissipative

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Definition

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

Parameters

Symbol	Default	Role
a	0.2	feedback strength
b	0.2	shift
c	5.7	fold control

State variables: x, y, z

Properties

Lyapunov spectrum

$+0.0714,\; 0,\; -5.39$
Sprott (2003), Chaos and Time-Series Analysis

Kaplan–Yorke dimension

$D_{KY} = 2.013$

Divergence ∇·f

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

Equilibria

2 equilibria
0 stable · 2 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Rossler()
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 (1976), Phys. Lett. A 57, 397-398
doi:10.1016/0375-9601(76)90101-8

BibTeX

@misc{rossler,
  title = {Rossler system},
  note = {Rössler (1976), Phys. Lett. A 57, 397-398},
  doi = {10.1016/0375-9601(76)90101-8}
}