VossDelay

Systems / DDEs / Delayed systems

Voss's anticipating-synchronization circuit — a scalar delay system with cubic feedback whose broad Lyapunov spectrum marks a high-dimensional delay attractor.

delay · DDE1 dimensionchaotic

phase portrait

Definition

\[ \begin{aligned} \dot{x} &= - \alpha x - 10.44 x^{3}{\left(t - \tau \right)} - 13.95 x^{2}{\left(t - \tau \right)} - 3.63 x{\left(t - \tau \right)} + 0.85 \end{aligned} \]

Parameters

Symbol	Default	Role
alpha	3.24	linear relaxation rate
tau	13.28	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

$\nabla\!\cdot f = - \alpha$
constant

Equilibria

TODO — fixed_points failed: NotImplementedError

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.VossDelay()
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

Voss (2002), Int. J. Bifurc. Chaos 12, 1619-1625

BibTeX

@misc{vossdelay,
  title = {VossDelay system},
  note = {Voss (2002), Int. J. Bifurc. Chaos 12, 1619-1625}
}