2014/10/10 by G. Contopoulos, Christos Efthymiopoulos, C. Efthymiopoulos +5
Physics and Astronomy · #Chaos control and synchronization #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Quantum chaos and dynamical systems #Scientific Research and Discoveries #nlin.CD
paper · pdf · doi:10.48550/arxiv.1410.2761
13 pages, 10 figures, published in the Proceedings of the 11th Hellenic Astronomical Conference (2013)
arxiv created 2014/10/10 · openalex publication_date 2014/10/10 · arxiv updated 2014/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a 2D conservative Hamiltonian system there is a formal integral Φ besides the energy H. This is not convergent near a stable periodic orbit, but it is convergent near an unstable periodic orbit. We explain this difference and we find the convergence radius along the asymptotic curves. In simple mappings this radius is infinite. This allows the theoretical calculation of the asymptotic curves and their intersections at homoclinic points. However in more complex mappings and in Hamiltonian systems the radius of convergence is in general finite and does not allow the theoretical calculation of any homoclinic point. Then we develop a method similar to analytic continuation, applicable in systems expressed in action-angle variables, that allows the calculation of the asymptotic curves to an arbitrary length. In this way we can study analytically the chaotic regions near the unstable periodic orbit and near its homoclinic points.