CellularNeuralNetwork¶
Systems / ODEs / Exotic systems
A three-cell continuous-time cellular neural network with piecewise-linear saturation — a hardware-realizable circuit chaos generator.
Definition¶
\[
\begin{aligned}
\dot{x} &= - 0.5 b \left(- \left|{y - 1}\right| + \left|{y + 1}\right|\right) - 0.5 b \left(- \left|{z - 1}\right| + \left|{z + 1}\right|\right) + 0.5 d \left(- \left|{x - 1}\right| + \left|{x + 1}\right|\right) - x \\
\dot{y} &= - 0.5 a \left(- \left|{z - 1}\right| + \left|{z + 1}\right|\right) - 0.5 b \left(- \left|{x - 1}\right| + \left|{x + 1}\right|\right) + 0.5 c \left(- \left|{y - 1}\right| + \left|{y + 1}\right|\right) - y \\
\dot{z} &= 0.5 a \left(- \left|{y - 1}\right| + \left|{y + 1}\right|\right) - 0.5 b \left(- \left|{x - 1}\right| + \left|{x + 1}\right|\right) - z - 0.5 \left|{z - 1}\right| + 0.5 \left|{z + 1}\right|
\end{aligned}
\]
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) |
State variables: x, y, z
Properties¶
Lyapunov spectrum
TODO — lyapunov_spectrum failed: AttributeError
Kaplan–Yorke dimension
TODO — requires a numeric Lyapunov spectrum
Divergence ∇·f
n/a — non-smooth right-hand side — divergence is piecewise (defined almost everywhere)
Equilibria
3 equilibria
0 stable · 3 unstable
0 stable · 3 unstable
Use it in TSDynamics¶
import tsdynamics as ts
system = ts.systems.CellularNeuralNetwork()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)
spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)
Reference¶
Arena, Caponetto, Fortuna & Porto (1998), Int. J. Bifurc. Chaos 8, 1527
doi:10.1142/s0218127498001170