Lissajous3D

Systems / ODEs / Oscillatory systems

A three-frequency Lissajous curve traced on a torus.

continuous · ODE3 dimensionsquasiperiodic

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
A	1	amplitude along the x axis
B	1	amplitude along the y axis
C	1	amplitude along the z axis
a	3	frequency along the x axis
b	2	frequency along the y axis
c	5	frequency along the z axis
delta_y	(1/2)*pi	phase shift of y relative to x
delta_z	(1/4)*pi	phase shift of z relative to x

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Lissajous3D()
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)