MackeyGlass

Systems / DDEs / Delayed systems

A single delayed feedback loop modeling blood-cell production — the canonical high-dimensional delay chaos.

delay · DDE1 dimensionchaotic

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Definition

\[ \begin{aligned} \dot{x} &= \frac{\beta x{\left(t - \tau \right)}}{x^{n}{\left(t - \tau \right)} + 1.0} - \gamma x \end{aligned} \]

Parameters

Symbol	Default	Role
beta	0.2	production gain
gamma	0.1	decay rate
tau	17	feedback delay (raise it to climb the period-doubling cascade into chaos)
n	10	Hill exponent

State variables: x

Properties

Lyapunov spectrum

≥ 1 positive exponent (literature — chaotic at tau = 17)

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

TODO — fixed_points failed: NotImplementedError

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.MackeyGlass()
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

Mackey & Glass (1977), Science 197, 287-289

BibTeX

@misc{mackeyglass,
  title = {MackeyGlass system},
  note = {Mackey & Glass (1977), Science 197, 287-289}
}