Lissajous2D¶

Systems / ODEs / Oscillatory systems

A planar Lissajous figure — the elementary two-frequency oscillation.

continuous · ODE2 dimensionsquasiperiodic

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= - A a \sin{\left(a t \right)} \\ \dot{y} &= - B b \sin{\left(b t + \delta \right)} \end{aligned} \]

Parameters¶

Symbol	Default	Role
`A`	`1`	amplitude along the x axis
`B`	`1`	amplitude along the y axis
`a`	`3`	frequency along the x axis
`b`	`2`	frequency along the y axis
`delta`	`(1/2)*pi`	phase shift of y relative to x

State variables: x, y

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 0$
constant

Equilibria

none found (no equilibria)

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Lissajous2D()
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)