IkedaDelay¶

Systems / DDEs / Delayed systems

A delayed optical-cavity model — sinusoidal nonlinearity with memory, chaotic for modest delay.

delay · DDE1 dimensionchaotic

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= - c x{\left(t \right)} - \mu \sin{\left(x_{0} - x{\left(t - \tau \right)} \right)} \end{aligned} \]

Parameters¶

Symbol	Default	Role
`c`	`1`	linear relaxation (damping) rate of the cavity field
`mu`	`-20`	feedback strength
`tau`	`2`	round-trip delay
`x0`	`0`	phase offset (detuning) of the nonlinear feedback

State variables: x

Properties¶

Lyapunov spectrum

TODO — DDE Lyapunov not computed at build time

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

TODO — fixed_points failed: NotImplementedError

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.IkedaDelay()
traj = system.run(final_time=500.0, dt=0.5)
ts.plot(traj)

# DDE Lyapunov uses the infinite-dimensional-history estimator:
spectrum = ts.analysis.lyapunov_spectrum(system, k=1, dt=0.5, ic=traj.y[-1])

Reference¶

Ikeda & Matsumoto (1987), Physica D 29, 223-235
doi:10.1016/0167-2789(87)90058-3

BibTeX

@misc{ikedadelay,
  title = {IkedaDelay system},
  note = {Ikeda & Matsumoto (1987), Physica D 29, 223-235},
  doi = {10.1016/0167-2789(87)90058-3}
}