ShimizuMorioka¶

Systems / ODEs / Oscillatory systems

A symmetric reduction of Lorenz-like dynamics near a homoclinic bifurcation.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

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

Parameters¶

Symbol	Default	Role
`a`	`0.85`	damping bifurcation parameter
`b`	`0.5`	dissipation bifurcation parameter

Properties¶

Lyapunov spectrum

$+0.02749,\; +0.005654,\; -1.383$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.024$

Divergence ∇·f

$\nabla\!\cdot f = - a - b$
constant

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Shimizu & Morioka (1980), Phys. Lett. A 76, 201-204

BibTeX

@misc{shimizumorioka,
  title = {ShimizuMorioka system},
  note = {Shimizu & Morioka (1980), Phys. Lett. A 76, 201-204}
}