Chua¶

Systems / ODEs / Chaotic attractors

The double-scroll attractor of Chua's circuit — the canonical electronic chaos generator.

continuous · ODE3 dimensionschaoticcircuit

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Definition¶

\[ \begin{aligned} \dot{x} &= \alpha \left(- m_{1} x - x + y - 0.5 \left(m_{0} - m_{1}\right) \left(- \left|{x - 1}\right| + \left|{x + 1}\right|\right)\right) \\ \dot{y} &= x - y + z \\ \dot{z} &= - \beta y \end{aligned} \]

Parameters¶

Symbol	Default	Role
`alpha`	`15.6`	dimensionless circuit parameter (capacitor/inductor ratio)
`beta`	`28`	dimensionless circuit parameter (capacitor/inductor ratio)
`m0`	`-1.14286`	inner slope of the piecewise-linear Chua diode
`m1`	`-0.71429`	outer slope of the piecewise-linear Chua diode

State variables: x, y, z

Properties¶

Lyapunov spectrum

≥ 1 positive exponent (literature — Matsumoto, Chua & Komuro (1985), IEEE Trans. Circuits Syst. 32, 798-818)

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

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Chua()
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¶

Matsumoto (1984), IEEE Trans. Circuits Syst. 31, 1055-1058
doi:10.1109/tcs.1984.1085459

BibTeX

@misc{chua,
  title = {Chua system},
  note = {Matsumoto (1984), IEEE Trans. Circuits Syst. 31, 1055-1058},
  doi = {10.1109/tcs.1984.1085459}
}