ShimizuMorioka¶

Systems / ODEs / Oscillatory systems

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

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

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

Parameters¶

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

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.ShimizuMorioka()
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¶

Shimizu & Morioka (1980), Phys. Lett. A 76, 201-204
doi:10.1016/0375-9601(80)90466-1

BibTeX

@misc{shimizumorioka,
  title = {ShimizuMorioka system},
  note = {Shimizu & Morioka (1980), Phys. Lett. A 76, 201-204},
  doi = {10.1016/0375-9601(80)90466-1}
}