IkedaDelay¶

Systems / DDEs / Delayed systems

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

delay · DDE1 dimensionchaotic

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Definition¶

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

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

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

Define it in TSDynamics¶

import tsdynamics as ts

sys = ts.systems.IkedaDelay()
traj = sys.integrate(final_time=500.0, dt=0.5)

# DDE Lyapunov uses the infinite-dimensional-history estimator:
exps = sys.lyapunov_spectrum(n_exp=1, dt=0.5, ic=traj.y[-1])

Reference¶

Ikeda & Matsumoto (1987), Physica D 29, 223-235

BibTeX

@misc{ikedadelay,
  title = {IkedaDelay system},
  note = {Ikeda & Matsumoto (1987), Physica D 29, 223-235}
}