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

\[ \begin{aligned} \dot{y_{0}} &= y_{1} \\ \dot{y_{1}} &= \gamma \cos{\left(y_{2} \right)} - \frac{\left(1 - \operatorname{sign}{\left(y_{0} \right)}\right) \left(\psi y_{1} \left|{y_{1}}\right| + y_{0}\right)}{2} - 1 \\ \dot{y_{2}} &= w \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.09433,\; -0.1139,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.829$

Divergence ∇·f

n/a — non-smooth right-hand side — divergence is piecewise (defined almost everywhere)

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Gilet & Bush (2009), J. Fluid Mech. 625, 167-203

BibTeX

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