ScrollDelay¶

Systems / DDEs / Delayed systems

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

delay · DDE1 dimensionchaotic

Interactive: drag to rotate

Definition¶

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

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

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.ScrollDelay()
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¶

Driver (1977), Ordinary and Delay Differential Equations, Springer
doi:10.1007/978-1-4684-9467-9_5

BibTeX

@misc{scrolldelay,
  title = {ScrollDelay system},
  note = {Driver (1977), Ordinary and Delay Differential Equations, Springer},
  doi = {10.1007/978-1-4684-9467-9_5}
}