Robinson

Systems / ODEs / Exotic systems

A Duffing-like double-well x–y oscillator coupled to a slow z mode driven by x², producing a Lorenz-type chaotic attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= y \\ \dot{y} &= - a y + b x^{2} y - v y z - 2 x^{3} + x \\ \dot{z} &= - c z + d x^{2} \end{aligned} \]

Parameters

Symbol	Default	Role
a	-0.42	linear damping of the oscillator velocity y
b	-1.1	coefficient of the x²·y nonlinear damping term
c	0.5	damping of the slow z mode
d	0.3	gain of the x² forcing on z
v	-1	coupling strength of the y·z feedback term

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a + b x^{2} - c - v z$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Robinson()
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)