2001/05/31 by M. Brack, Minesh P. Mehta, M. Mehta +2 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Chaos control and synchronization #Quantum chaos and dynamical systems #math-ph #math.MP #nlin.CD
paper · pdf · doi:10.1088/0305-4470/34/40/301
published as Journal of Physics A 34, 8199 (2001) · LaTeX2e, 22 pages, 14 figures. Version 3: final form of paper, accepted by J. Phys. A. Changes in Table 2; new reference [25]; name of bifurcations "touch-and-go" replaced by "island-chain"
arxiv created 2001/08/30 · openalex publication_date 2001/10/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We investigate cascades of isochronous pitchfork bifurcations of straight-line librating orbits in some two-dimensional Hamiltonian systems with mixed phase space. We show that the new bifurcated orbits, which are responsible for the onset of chaos, are given analytically by the periodic solutions of the Lamé equation as classified in 1940 by Ince. In Hamiltonians with C 2 v symmetry, they occur alternatingly as Lamé functions of period 2 K and 4 K , respectively, where 4 K is the period of the Jacobi elliptic function appearing in the Lamé equation. We also show that the two pairs of orbits created at period-doubling bifurcations of island-chain type are given by two different linear combinations of algebraic Lamé functions with period 8 K .