ScrollDelay

Systems / DDEs / Delayed systems

A delayed feedback circuit producing multi-scroll attractors in the embedding.

delay · DDE1 dimensionchaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= - \alpha y_{0}{\left(t - \tau \right)} + \beta \tanh{\left(10 y_{0}{\left(t - \tau \right)} \right)} \end{aligned} \]

Parameters

Symbol	Default	Role
alpha	0.2	linear decay rate of the delayed state
beta	0.2	gain of the saturating tanh feedback
tau	10	feedback delay

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.ScrollDelay()
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

Driver (1977), Ordinary and Delay Differential Equations, Springer

BibTeX

@misc{scrolldelay,
  title = {ScrollDelay system},
  note = {Driver (1977), Ordinary and Delay Differential Equations, Springer}
}