Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems

Abstract

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.

Semiclassical trace formulas for pitchfork bifurcation sequences

“…Due to the behaviour of Tr M ξ near the bifurcation, a generic perioddoubling bifurcation is often called a pitchfork bifurcation. The case of interest here is a sequence of two such nongeneric pitchfork bifurcations which can arise successively from the same central periodic orbit in systems with discrete symmetries such as studied in [18,19]. For this scenario we propose the new normal form…”

“…The orbit B is unstable for all energies and the orbit C stays stable for energies below e = 0.8922 where it becomes unstable due to a generic period-doubling bifurcation. The bifurcation cascades of the A orbit and the orbits generated by them have been studied in detail in [18,19]; we adapt the names of the orbits given in these references, whereby the subscripts of the orbit names denote their Maslov indices. The A orbit undergoes its first isochronous pitchfork bifurcation at an energy e 1 = 0.969309 and the second one at e 2 = 0.986709.…”

“…Explicit semiclassical trace formulae have been given for various systems with continous symmetries [3,5,6,7], for symmetry breaking through the destruction of rational tori [8,9,10,11,12], and for isolated bifurcations [13,14,15]. However more complicated bifurcation scenarios which usually occur in realistic physical systems and, in particular, bifurcation cascades [16,17,18,19] still constitute one of the most serious problems of the semiclassical theory.…”

“…In fact, it may constitute the beginning of a bifurcation cascade in which this sequence is repeated infinitely often. Such a cascade can form a geometric progression reminiscent of the Feigenbaum scenario [26] (although there the bifurcations are generically period doubling), and the new periodic orbits born at the bifurcations may exhibit self-similarity properties [18,19]. Bifurcation cascades are frequently found in physical systems with discrete symmetries and mixed classical dynamics [16,17], so that a semiclassical approach to those situations would seem to have been required long ago.…”

Periodic orbit theory for the Hénon-Heiles system in the continuum region

“…The shortest periodic orbits of the classical HH system (12) were obtained using a numerical Newton-Raphson algorithm [39]. They have already been extensively studied in earlier papers [40][41][42][43][44]. We use here the nomenclature introduced in [44], where the Maslov indices σ ξ appear as subscripts in the symbols (B 4 , R 5 , L 6 , etc.)…”