# Bifurcation between two oscillations

1. Oct 5, 2015

### onkel_tuca

Hello world!

I've done a few simulations of an emulsion droplet which is actuated by a laser beam. The droplet starts to move due to the laser light. I don't want to talk too much about the physics behind this but more discuss the nonlinear dynamics of the trajectories. Depending on a parameter "1/kappa", the droplet dynamics is either

(1) damped leading to a stop of the drop
(2) oscillating around the beam
(3) oscillating around the beam, but then changing its direction
(4) the droplet shoots completely out of the laser beam and stops.

From my understanding, the first bifurcation between (1) and (2) is a typical Hopf bifurcation. See attached plot. There you see four phase-space plots (velocity vs. displacement) and a plot of the amplitude A and wavenumber \nu(=1/ wavelength) of the oscillations.

However I'm not sure if one can classify the second bifurcation between (2) and (3). In case (3) the dynamics is first along oscillation (2), then the droplet changes direction and increases its amplitude A and wavenumber \nu(=1/ wavelength of oscillation) and stays on the outer orbit.

Thus my question is: Is there a name for a bifurcation between two (very) different oscillations?

Cheers!

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2. Oct 5, 2015

### Krylov

I agree, it seems to be a supercritical Andronov-Hopf, at least this is what your simulation suggests.

It seems to me from your plot that the wave number $\nu$ in case (3) has actually decreased? Could it be that the transition (2) $\rightarrow$ (3) indicates a period doubling (flip) bifurcation while the transition (3) $\rightarrow$ (4) is a fold bifurcation of cycles?

EDIT: You might want to look into Kuznetsov's book, "Elements of Applied Bifurcation Theory". The fourth chapter could be of interest. What is the mathematical form of your model? An ODE?

3. Oct 11, 2015

### onkel_tuca

Hey Krylov,

thanks for your answer. I missed the email about it. The first bifurcation is actually subcritical (the position of the bifurcation depends a little bit on the initial amplitude, i.e. there's a small overlap of the damped (1) and the osc. case (2)).

The mathematical form of the model is rather complex, it's a nonlinear PDE (react.-diff.-adv. eqn) on a sphere coupled to a Stokes equation for the flow field...

There's no period flipping between (2) and (3). Also I've looked up "fold bifurcation of cycles" in Strogatz and I think that's something different than the bifurcation from (3) to (4). Just for fun I'll add a video of the four simulations. There you can see the four cases from top to bottom.

To be honest I'm not even sure if I can call that a "phase space" since trajectories cross each other. In reality my phase space is N>>1 dimensional, and I'm just projecting onto 2 dimensions. Is it still save to do these kind of analyses for a "projected phase space"?