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Zbl 1051.37027
Brack, M.; Mehta, M.; Tanaka, K.
Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. (English)
[J] J. Phys. A, Math. Gen. 34, No. 40, 8199-8220 (2001). ISSN 0305-4470

Summary: 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 $C_{2v}$ symmetry, they occur alternatingly as Lamé functions of period $2K$ and $4K$, respectively, where $4K$ 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 $8K$.

MSC 2000:: *37J20 Bifurcation problems
33E10 Spheroidal wave functions, etc.
70H05 Hamilton's equations
70K50 Transition to stochasticity (general mechanics)
82B05 Classical equilibrium statistical mechanics (general)

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