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Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. (English) Zbl 1051.37027
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:
 37J20 Bifurcation problems (finite-dimensional Hamiltonian etc. systems) 33E10 Lamé, Mathieu, and spheroidal wave functions 70H05 Hamilton’s equations 70K50 Transition to stochasticity (general mechanics) 82B05 Classical equilibrium statistical mechanics (general)
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