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¶

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

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

Properties¶

Lyapunov spectrum

$+0.004119,\; +0.06717,\; -0.9513$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.075$

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Kennedy (1994), IEEE Trans. Circuits Syst. I 41, 771-774

BibTeX

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