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

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

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

Properties

Lyapunov spectrum

$+0.005799,\; -0.03183,\; -3.278$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1.182$

Divergence ∇·f

$\nabla\!\cdot f = - a + b y_{0}^{2} - c - v y_{2}$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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