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In the theory of dynamical systems, a bailout embedding is a system defined as

${\displaystyle {\begin{aligned}&{\frac {d}{dt}}(u-f(x))=-k(x)(u-f(x)),\\&{\frac {dx}{dt}}=u.\end{aligned}}}$

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.

References

1. Cartwright, Julyan H. E.; Magnasco, Marcelo O.; Piro, Oreste (2002-04-03). "Bailout embeddings, targeting of invariant tori, and the control of Hamiltonian chaos". Physical Review E. 65 (4). American Physical Society (APS): 045203(R). arXiv:nlin/0111005. Bibcode:2002PhRvE..65d5203C. doi:10.1103/physreve.65.045203. ISSN 1063-651X. PMID 12005907. S2CID 23498762.
2. Tuval, Idan; Piro, Oreste (2003). "Bailout Embedding as a Blowout Bifurcation". Progress of Theoretical Physics Supplement. 150. Oxford University Press (OUP): 465–468. Bibcode:2003PThPS.150..465T. doi:10.1143/ptps.150.465. hdl:10261/15339. ISSN 0375-9687.
3. Shan, Zhang; Shi-Ping, Yang; Hu, Liu (2006-04-28). "Targeting of Kolmogorov–Arnold–Moser Orbits by the Bailout Embedding Method in Two Coupled Standard Maps". Chinese Physics Letters. 23 (5). IOP Publishing: 1114–1117. Bibcode:2006ChPhL..23.1114Z. doi:10.1088/0256-307x/23/5/014. ISSN 0256-307X. S2CID 250847203.
4. Thyagu, N. Nirmal; Gupte, Neelima (2007-10-22). "Clustering, chaos, and crisis in a bailout embedding map". Physical Review E. 76 (4): 046218. arXiv:0707.3102. Bibcode:2007PhRvE..76d6218T. doi:10.1103/physreve.76.046218. ISSN 1539-3755. PMID 17995093. S2CID 1801240.

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Physical systems	Chua's circuit Convection Double pendulum Elastic pendulum FPUT problem Hénon–Heiles system Kicked rotator Multiscroll attractor Population dynamics Swinging Atwood's machine Tilt-A-Whirl Weather
Chaos theorists	Michael Berry Rufus Bowen Mary Cartwright Chen Guanrong Leon O. Chua Mitchell Feigenbaum Peter Grassberger Celso Grebogi Martin Gutzwiller Brosl Hasslacher Michel Hénon Svetlana Jitomirskaya Bryna Kra Edward Norton Lorenz Aleksandr Lyapunov Benoît Mandelbrot Hee Oh Edward Ott Henri Poincaré Itamar Procaccia Mary Rees Otto Rössler David Ruelle Caroline Series Yakov Sinai Oleksandr Mykolayovych Sharkovsky Nina Snaith Floris Takens Audrey Terras Mary Tsingou Marcelo Viana Amie Wilkinson James A. Yorke Lai-Sang Young
Related articles	Butterfly effect Complexity Edge of chaos Predictability Santa Fe Institute

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