Analytic approach to bifurcation cascades in a class of generalized Hénon-Heiles potentials

2008/04/01 by S. N. Fedotkin, A. G. Magner, M. Brack
Computer Science · Mathematics · Physics and Astronomy · #Bifurcation #Chaos control and synchronization #Chaotic #Class (philosophy) #Classical mechanics #Geometry #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Orbit (dynamics) #Periodic orbits #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Saddle #Saddle point #Stability (learning theory) #Statistical physics #nlin.CD

paper · pdf · doi:10.1103/physreve.77.066219

LaTeX revtex4, 38 pages, 7 PostScript figures, 2 tables

arxiv created 2008/04/01 · openalex publication_date 2008/06/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We investigate the bifurcation cascades of a linear librational orbit in a generalized class of Hénon-Heiles potentials. The stability traces of the new orbits created at its bifurcations are found numerically to intersect linearly at the saddle energy (e=1) , forming what we term the "Hénon-Heiles fans." In the limit close to the saddle energy (e-->1) , where the dynamics is nearly chaotic, we derive analytical asymptotic expressions for the stability traces of both types of orbits and confirm the numerically determined properties of the generalized Hénon-Heiles fans. As a bonus of our results, we obtain analytical approximations for the bifurcation energies e_n which become asymptotically exact for e_n-->1 .

Citations

• Periodic Orbits and Classical Quantization Conditions
• Bifurcations of periodic orbits and uniform approximations
• Transcritical bifurcations in nonintegrable Hamiltonian systems
• Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems
• Quantitative universality for a class of nonlinear transformations