Lissajous2D

Systems / ODEs / Oscillatory systems

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

continuous · ODE2 dimensionsquasiperiodic

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Definition

\[ \begin{aligned} \dot{y_{0}} &= - A a \sin{\left(a t \right)} \\ \dot{y_{1}} &= - B b \sin{\left(b t + \delta \right)} \end{aligned} \]

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

Properties

Lyapunov spectrum

$0,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2$

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Lissajous2D()
traj = sys.integrate(final_time=100.0, dt=0.01)

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