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.
We investigate the spatiotemporal dynamics of two dimensional coupled map lattices, in the strong coupling phase, evolving under updating rules incorporating varying degrees of asynchronicity. Interestingly, we observe that parallel updates never allow synchronization among the sites, while asynchroncity has the effect of opening up windows in parameter space where the synchronized dynamics gains stability. As asynchronicity increases, the parameter range supporting synchronization gets rapidly wider. Detailed numerics, including bifurcation diagrams and patterns formed en route to synchronization, are reported. We also attempt a mean-field analysis of the system in order to try and account for the stability of the spatiotemporal fixed point under asynchronous updates. (c) 2000 American Institute of Physics.
We present an analytical calculation of periodic orbits in the homogeneous quartic oscillator potential. Exploiting the properties of the periodic Lamé functions that describe the orbits bifurcated from the fundamental linear orbit in the vicinity of the bifurcation points, we use perturbation theory to obtain their evolution away from the bifurcation points. As an application, we derive an analytical semiclassical trace formula for the density of states in the separable case, using a uniform approximation for the pitchfork bifurcations occurring there, which allows for full semiclassical quantization. For the non-integrable situations, we show that the uniform contribution of the bifurcating period-one orbits to the coarse-grained density of states competes with that of the shortest isolated orbits, but decreases with increasing chaoticity parameter α.
We calculate the geometric phase associated with the time evolution of the wave function of a Bose-Einstein condensate system in a double-well trap by using a model for tunneling between the wells. For a cyclic evolution, this phase is shown to be half the solid angle subtended by the evolution of a unit vector whose z component and azimuthal angle are given, respectively, by the population difference and phase difference between the two condensates. For a non-cyclic evolution, an additional phase term arises. We show that the geometric phase can also be obtained by mapping the tunneling equations on to the equations of a space curve. The importance of a geometric phase in the context of some recent experiments is pointed out.
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