Laser¶

Systems / ODEs / Physical systems

A single-mode laser (Haken) — the Lorenz equations in optical disguise.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

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

Parameters¶

Symbol	Default	Role
`a`	`10`	linear coupling / decay rate between x and y
`b`	`1`	strength of the cubic y*z^2 coupling driving x
`c`	`5`	linear coupling coefficient driving y
`d`	`-1`	cubic (x*z^2) coupling coefficient driving y
`h`	`-5`	linear damping of z
`k`	`-6`	strength of the quadratic x^2 feedback into z

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a + h$
constant

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Laser()
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¶

Abooee, Yaghini-Bonabi & Jahed-Motlagh (2013), Commun. Nonlinear Sci. Numer. Simul. 18, 1235-1245
doi:10.1016/j.cnsns.2012.08.036

BibTeX

@misc{laser,
  title = {Laser system},
  note = {Abooee, Yaghini-Bonabi & Jahed-Motlagh (2013), Commun. Nonlinear Sci. Numer. Simul. 18, 1235-1245},
  doi = {10.1016/j.cnsns.2012.08.036}
}