Laser

Systems / ODEs / Physical systems

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

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= - a y_{0} + a y_{1} + b y_{1} y_{2}^{2} \\ \dot{y_{1}} &= c y_{0} + d y_{0} y_{2}^{2} \\ \dot{y_{2}} &= h y_{2} + k y_{0}^{2} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.5705,\; -0.006759,\; -15.56$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.036$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Abooee, Yaghini-Bonabi & Jahed-Motlagh (2013), Commun. Nonlinear Sci. Numer. Simul. 18, 1235-1245

BibTeX

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