Bailout embedding explained
In the theory of dynamical systems, a bailout embedding is a system defined as[1] [2] [3] [4]
(u-f(x))=-k(x)(u-f(x)),\\[8pt]
&
=u.
\end{align}
Here the function
k(
x) < 0 on a
set of unwanted
orbits; otherwise
k(
x) > 0. The
trajectories of the full system of a bailout embedding
bail out—that is, detach—from the
embedding, into a larger space, in which they move around. If, after some time these orbits arrive at a
stable neighbourhood of the embedding,
k(
x) > 0, they collapse once more onto the embedding; that is, onto the original
dynamics. The bailout embedding forms in this way an enlarged version of the
dynamical system, one in which particular sets of orbits are cut from the asymptotic or
limit set, while maintaining the dynamics of a different set of orbits—the wanted set—as
attractors of the larger dynamical system. With a choice of
k(
x) = −(
γ + ∇
f), these dynamics are seen to detach from unstable regions such as
saddle points in
conservative systems.
One important application of the bailout embedding concept is to divergence-free flows; the most important class of these are Hamiltonian systems.
Notes and References
- Cartwright . Julyan H. E. . Julyan Cartwright. Magnasco . Marcelo O. . Piro . Oreste . Bailout embeddings, targeting of invariant tori, and the control of Hamiltonian chaos . Physical Review E . American Physical Society (APS) . 65 . 4 . 2002-04-03 . 1063-651X . 10.1103/physreve.65.045203 . 045203(R). 12005907 . nlin/0111005. 2002PhRvE..65d5203C . 23498762 .
- Tuval . Idan . Piro . Oreste . Bailout Embedding as a Blowout Bifurcation . Progress of Theoretical Physics Supplement . Oxford University Press (OUP) . 150 . 2003 . 0375-9687 . 10.1143/ptps.150.465 . 465–468. 2003PThPS.150..465T . free. 10261/15339 . free .
- Shan . Zhang . Shi-Ping . Yang . Hu . Liu . Targeting of Kolmogorov–Arnold–Moser Orbits by the Bailout Embedding Method in Two Coupled Standard Maps . Chinese Physics Letters . IOP Publishing . 23 . 5 . 2006-04-28 . 0256-307X . 10.1088/0256-307x/23/5/014 . 1114–1117. 2006ChPhL..23.1114Z . 250847203 .
- Thyagu . N. Nirmal . Gupte . Neelima . Clustering, chaos, and crisis in a bailout embedding map . Physical Review E . 76 . 4 . 2007-10-22 . 1539-3755 . 10.1103/physreve.76.046218 . 046218. 17995093 . 0707.3102. 2007PhRvE..76d6218T . 1801240 .
