Bogdanov¶

Systems / Maps / Exotic maps

The discrete-time normal form of the Bogdanov-Takens double-zero bifurcation — invariant circles, Arnold tongues, and chaos near a cusp.

discrete · map2 dimensionschaotic

Definition¶

\[ \begin{aligned} x' &= k x \left(x - 1\right) + \mu x y + x + y \left(eps + 1\right) \\ y' &= k x \left(x - 1\right) + \mu x y + y \left(eps + 1\right) \end{aligned} \]

Parameters¶

Symbol	Default	Role
`eps`	`0`	linear damping/detuning (the Hopf parameter)
`k`	`1.2`	strength of the quadratic x(x-1) nonlinearity
`mu`	`0`	coefficient of the mixed x*y term

State variables: x, y

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

n/a — discrete map — flow divergence undefined (per-step contraction is |det J|)

Equilibria

2 fixed points
0 stable · 2 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Bogdanov()
traj = system.run(steps=10_000)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference¶

Bogdanov (1981), Selecta Math. Soviet. 1, 389-421

BibTeX

@misc{bogdanov,
  title = {Bogdanov system},
  note = {Bogdanov (1981), Selecta Math. Soviet. 1, 389-421}
}