CellularNeuralNetwork

Systems / ODEs / Exotic systems

A three-cell continuous-time cellular neural network with piecewise-linear saturation — a hardware-realizable circuit chaos generator.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= - 0.5 b \left(- \left|{y_{1} - 1}\right| + \left|{y_{1} + 1}\right|\right) - 0.5 b \left(- \left|{y_{2} - 1}\right| + \left|{y_{2} + 1}\right|\right) + 0.5 d \left(- \left|{y_{0} - 1}\right| + \left|{y_{0} + 1}\right|\right) - y_{0} \\ \dot{y_{1}} &= - 0.5 a \left(- \left|{y_{2} - 1}\right| + \left|{y_{2} + 1}\right|\right) - 0.5 b \left(- \left|{y_{0} - 1}\right| + \left|{y_{0} + 1}\right|\right) + 0.5 c \left(- \left|{y_{1} - 1}\right| + \left|{y_{1} + 1}\right|\right) - y_{1} \\ \dot{y_{2}} &= 0.5 a \left(- \left|{y_{1} - 1}\right| + \left|{y_{1} + 1}\right|\right) - 0.5 b \left(- \left|{y_{0} - 1}\right| + \left|{y_{0} + 1}\right|\right) - y_{2} - 0.5 \left|{y_{2} - 1}\right| + 0.5 \left|{y_{2} + 1}\right| \end{aligned} \]

Parameters

Symbol	Default	Role
a	4.4	synaptic coupling weight (CNN template feedback)
b	3.21	synaptic coupling weight (CNN template feedback)
c	1.1	synaptic coupling weight (CNN template feedback)
d	1.24	synaptic coupling weight (CNN template feedback)

Properties

Lyapunov spectrum

$+0.08789,\; +0.0186,\; -0.6136$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.174$

Divergence ∇·f

n/a — non-smooth right-hand side — divergence is piecewise (defined almost everywhere)

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.CellularNeuralNetwork()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Arena, Caponetto, Fortuna & Porto (1998), Int. J. Bifurc. Chaos 8, 1527

BibTeX

@misc{cellularneuralnetwork,
  title = {CellularNeuralNetwork system},
  note = {Arena, Caponetto, Fortuna & Porto (1998), Int. J. Bifurc. Chaos 8, 1527}
}