Arxiv:Nlin.CD/0111005 V1 2 Nov 2001 Ple Namical Tems: on Asymptotically While Islands Lands W to the Apply Class
Bailout Embeddings, Targeting of KAM Orbits, and the Control of Hamiltonian Chaos 1, 2, 3, Julyan H. E. Cartwright, ∗ Marcelo O. Magnasco, y and Oreste Piro z 1Laboratorio de Estudios Cristalograficos,´ CSIC, E-18071 Granada, Spain 2Mathematical Physics Lab, Rockefeller University, Box 212, 1230 York Avenue, NY 10021 3Institut Mediterrani d’Estudis Avanc¸ats, CSIC–UIB, E-07071 Palma de Mallorca, Spain (Dated: version of December 11, 2001) We present a novel technique, which we term bailout embedding, that can be used to target orbits having par- ticular properties out of all orbits in a flow or map. We explicitly construct a bailout embedding for Hamiltonian systems so as to target KAM orbits. We show how the bailout dynamics is able to lock onto extremely small KAM islands in an ergodic sea. PACS numbers: 05.45.Gg Control of chaos in nonlinear dynamical systems has been a parameter taking the system away from integrability is in- achieved by applying small perturbations that effectively creased, these tori break and give rise to chaotic regions in a change the dynamics of the system around the region — typ- precise sequence; for any particular value of this parameter ically a periodic orbit — that one wishes to stabilize [1]. in a neighborhood of the integrable case, there are surviving This method has been successful in dissipative systems, but tori. The problem with finding them is that, the dynamics its extensions to control and targeting in Hamiltonian sys- being volume-preserving, merely evolving trajectories either tems [2, 3, 4, 5, 6, 7, 8, 9, 10] have met various difficulties forwards or backwards does not give us convergence onto tori not present in the dissipative case: the absence of attracting [13], and since for large values of the nonlinearity, they cover sets, for instance, makes it hard to stabilize anything.
[Show full text]