PiecewiseCircuit¶

Systems / DDEs / Delayed systems

A delayed piecewise-linear circuit — chaos from a saturating nonlinearity with memory.

delay · DDE1 dimensionchaotic

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= - \alpha x{\left(t - \tau \right)} + \beta \left(\frac{3 x{\left(t - \tau \right)}}{c} - \frac{x^{3}{\left(t - \tau \right)}}{c^{3}}\right) \end{aligned} \]

Parameters¶

Symbol	Default	Role
`alpha`	`1`	linear decay rate of the delayed state
`beta`	`1`	gain of the cubic feedback term
`c`	`2.24`	scale of the cubic nonlinearity (feedback hump width)
`tau`	`4.9`	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.PiecewiseCircuit()
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¶

Tamasevicius, Mykolaitis & Bumeliene (2006), Electron. Lett. 42, 13
doi:10.1049/el:20061245

BibTeX

@misc{piecewisecircuit,
  title = {PiecewiseCircuit system},
  note = {Tamasevicius, Mykolaitis & Bumeliene (2006), Electron. Lett. 42, 13},
  doi = {10.1049/el:20061245}
}