Bailout embeddings, targeting of invariant tori, and the control of Hamiltonian chaos

Abstract

We introduce a technique, which we term bailout embedding, that can be used to target orbits having particular properties out of all orbits in a flow or map. We explicitly construct a bailout embedding for Hamiltonian systems so as to target invariant tori. We show how the bailout dynamics are able to lock onto extremely small regular islands in a chaotic sea.

Noise-Induced Order Out of Chaos by Bailout Embedding

“…Bailout embeddings constitute a subclass of embeddings with much utility for obtaining order from chaotic systems. Here, as in previous work, we have provided examples of the use of bailout embeddings for finding regularity -quasiperiodic orbits -in both Hamiltonian maps [3,4] and flows [1], and in the more general setting of volume-preserving maps [5]. We have also demonstrated that the addition of a small amount of noise enhances the bailout process in a nontrivial fashion [4].…”

“…Thus the embedding trajectories have detached from the actual trajectories, and the approximations given above break down. Detachment is the process that was first envisioned as being characteristic of bailout embeddings [1,3]. However, by employing noise in the embedding and carefully controlling its use, we can see the process that occurs before detachment.…”

Controlling synchronization in coupled area-preserving maps using stickiness

“…10). The classical problem of targeting invariant structures in Hamiltonian system [50,51] may be revisited if we understand the mechanism in detail and adapt it appropriately. As far as experimental situations are concerned, the stickiness is a known phenomenon with interesting consequences in a variety of systems such as intramolecular energy redistribution [52], microwave ionization of Rydberg atoms [53] etc.…”

The fluid mechanics of poohsticks

“…In 2000 [16], together with our colleagues Armando Babiano and Antonello Provenzale, we pointed out that from the Maxey-Riley equations one can see that even in the most favourable case of a small, neutrally buoyant particle, the particle will not always slavishly follow the fluid flow, but will on occasion bail out from following the flow and instead follow its own course for a while before rejoining the fluid flow (we subsequently studied this type of dynamical system, which is interesting in its own right, and for which we coined the term bailout embedding, with our colleague Marcelo Magnasco [17][18][19]). Larger particles, non-spherical particles, and particles heavier or lighter than the surrounding fluid are even less inclined to follow perfectly the fluid flow [20].…”