NDLI: Systematic Computer-Assisted Proof of Branches of Stable Elliptic Periodic Orbits and Surrounding Invariant Tori

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Wilczak, Daniel Barrio, Roberto
Copyright Year	2017
Abstract	We present a concurrent algorithm for rigorous validation of the existence of continuous branches of stable elliptic fixed points for area-preserving planar maps. The method utilizes a classical theorem of Siegel and Moser combined with computed-assisted estimation of higher order derivatives of maps, continuation along the parameter range, and concurrent scheduling of tasks. We apply the algorithm to certain exemplary Poincaré maps coming from reversible or Hamiltonian systems: the periodically forced pendulum equations, the Michelson system, and the Hénon--Heiles Hamiltonian. Moreover, our algorithm provides at once a computer-assisted proof of the existence of wide branches of stable elliptic periodic solutions and the existence of invariant tori surrounding them.
Sponsorship	Narodowe Centrum Nauki. Spanish Research Project
Starting Page	1618
Ending Page	1649
Page Count	32
File Format	PDF
DOI	10.1137/17M1113254
e-ISSN	15360040
Issue Number	3
Volume Number	16
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-08-22
Access Restriction	Subscribed
Subject Keyword	computer-assisted proof normal forms Stability Simulation Numerical investigation of stability of solutions resonances stable elliptic periodic orbit continuation invariant tori
Content Type	Text
Resource Type	Article

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