Tent¶

Systems / Maps / Geometric maps

The piecewise-linear tent map — exact, fully chaotic, conjugate to the binary shift.

discrete · map1 dimensionchaotic

Tent return map · bifurcation diagram — return map · bifurcation diagram

Definition¶

\[ \begin{aligned} x' &= \mu \left(1 - 2 \left|{x - 0.5}\right|\right) \end{aligned} \]

Parameters¶

Symbol	Default	Role
`mu`	`0.95`	peak height (branch slope magnitude 2*mu)

State variables: x

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

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

Reference¶

Classical map; see e.g. Strogatz, Nonlinear Dynamics and Chaos

BibTeX

@misc{tent,
  title = {Tent system},
  note = {Classical map; see e.g. Strogatz, Nonlinear Dynamics and Chaos}
}