Colpitts¶

Systems / ODEs / Physical systems

The Colpitts LC oscillator driven into chaos — a real electronic circuit model.

continuous · ODE3 dimensionschaoticcircuit

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= \frac{a \left(1 - \operatorname{sign}{\left(- e + z + 1 \right)}\right) \left(- e + z + 1\right)}{2} + y \\ \dot{y} &= - b y + c - x - z \\ \dot{z} &= - d z + y \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`30`	loop-gain / nonlinearity strength of the transistor term
`b`	`0.8`	damping (loss) coefficient on the voltage y
`c`	`20`	bias / supply drive term
`d`	`0.08`	inductor-branch loss coefficient on z
`e`	`10`	transistor turn-on (pinch-off) threshold

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - b - d$
constant

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Colpitts()
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¶

Kennedy (1994), IEEE Trans. Circuits Syst. I 41, 771-774
doi:10.1109/81.331536

BibTeX

@misc{colpitts,
  title = {Colpitts system},
  note = {Kennedy (1994), IEEE Trans. Circuits Syst. I 41, 771-774},
  doi = {10.1109/81.331536}
}