FluidTrampoline

Systems / ODEs / Physical systems

A droplet bouncing on a vibrated soap film that acts as a nonlinear spring — increasing forcing drives simple, multi-periodic, then chaotic bouncing.

continuous · ODE3 dimensionsforced, chaotic

phase portrait

Definition

\[ \begin{aligned} \dot{x} &= y \\ \dot{y} &= \gamma \cos{\left(th \right)} - \frac{\left(1 - \operatorname{sign}{\left(x \right)}\right) \left(\psi y \left|{y}\right| + x\right)}{2} - 1 \\ \dot{th} &= w \end{aligned} \]

Parameters

Symbol	Default	Role
gamma	1.82	dimensionless forcing acceleration amplitude
psi	0.01019	quadratic-drag energy-loss coefficient of film contact
w	1.21	dimensionless forcing angular frequency

State variables: x, y, th

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

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

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.FluidTrampoline()
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

Gilet & Bush (2009), J. Fluid Mech. 625, 167-203
doi:10.1017/s0022112008005442

BibTeX

@misc{fluidtrampoline,
  title = {FluidTrampoline system},
  note = {Gilet & Bush (2009), J. Fluid Mech. 625, 167-203},
  doi = {10.1017/s0022112008005442}
}