Discrete chaotic states of a Bose-Einstein condensate

We find the different spatial chaos in a one-dimensional attractive Bose-Einstein condensate interacting with a Gaussian-like laser barrier and perturbed by a weak optical lattice. For the low laser barrier the chaotic regions of parameters are demonstrated and the chaotic and regular states are illustrated numerically. In the high barrier case, the bounded perturbed solutions which describe a set of discrete chaotic states are constructed for the discrete barrier heights and magic numbers of condensed atoms. The chaotic density profiles are exhibited numerically for the lowest quantum number, and the analytically bounded but numerically unbounded Gaussian-like configurations are confirmed. It is shown that the chaotic wave packets can be controlled experimentally by adjusting the laser barrier potential.Comment: 7 pages, 5 figure

Multiple Transitions to Chaos in a Damped Parametrically Forced Pendulum

We study bifurcations associated with stability of the lowest stationary point (SP) of a damped parametrically forced pendulum by varying $\omega_0$ (the natural frequency of the pendulum) and $A$ (the amplitude of the external driving force). As $A$ is increased, the SP will restabilize after its instability, destabilize again, and so {\it ad infinitum} for any given $\omega_0$. Its destabilizations (restabilizations) occur via alternating supercritical (subcritical) period-doubling bifurcations (PDB's) and pitchfork bifurcations, except the first destabilization at which a supercritical or subcritical bifurcation takes place depending on the value of $\omega_0$. For each case of the supercritical destabilizations, an infinite sequence of PDB's follows and leads to chaos. Consequently, an infinite series of period-doubling transitions to chaos appears with increasing $A$. The critical behaviors at the transition points are also discussed.Comment: 20 pages + 7 figures (available upon request), RevTex 3.

Period p-tuplings in coupled maps

We study the critical behavior (CB) of all period $p$-tuplings $(p \!=\!2,3,4,\dots)$ in $N$ $(N \!=\! 2,3,4,\dots)$ symmetrically coupled one-dimensional maps. We first investigate the CB for the $N=2$ case of two coupled maps, using a renormalization method. Three (five) kinds of fixed points of the renormalization transformation and their relevant ``coupling eigenvalues'' associated with coupling perturbations are found in the case of even (odd) $p$. We next study the CB for the linear- and nonlinear-coupling cases (a coupling is called linear or nonlinear according to its leading term), and confirm the renormalization results. Both the structure of the critical set (set of the critical points) and the CB vary according as the coupling is linear or nonlinear. Finally, the results of the two coupled maps are extended to many coupled maps with $N \geq 3$, in which the CB depends on the range of coupling.Comment: RevTeX, 30 figures available upon reques