Isochrons are foliations of phase space that extend the notion of phase of a stable periodic orbit to the basin of attraction of this periodic orbit. Each point in the basin of attraction lies on only one isochron and two points on the same isochron converge to the periodic orbit with the same phase. These properties allow one to define so-called phase models that reduce the dimension of an osc...

In this paper we study periodic orbit bifurcation sequences in a system of two coupled Morse oscillators. Time-reversal symmetry is exploited to determine periodic orbits by iteration of symmetry lines. The permutational representation of Tsuchiya and Jaffe is employed to analyze periodic orbit configurations on the symmetry lines. Local pruning rules are formulated, and a global analysis of po...

– Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose a method for semiclassical quantization based upon the Padé approximant to the periodic orbit sums. The Padé approximant allows the re-summation of the typically exponentially divergent periodic orbit terms. The technique does not depend on the existence of a symbolic dynam...

We study almost periodic orbits of quantum systems and prove that for periodic time-dependent Hamiltonians an orbit is almost periodic if, and only if, it is precompact. In the case of quasiperiodic time-dependence we present an example of a precompact orbit that is not almost periodic. Finally we discuss some simple conditions assuring dynamical stability for nonautonomous quantum system.

A concept of orbit spectral density for a one-dimensional iterated function is presented. To compute orbit spectral density, a method of extracting low-order periodic orbits from the dynamical system defined by the iterated function is first used. All points of the dynamics are then partitioned among the periodic orbits according to a distance measure. Partition sizes estimate the density of tr...

A concept of phase asymptotic semiflow is defined. It is shown that any Lagrange stable orbit at which the semiflow is phase asymptotic limits to a stable periodic orbit. A Lagrange stable solution of a C 1 differential equation is considered. When the second compound of the variational equation with respect to this solution is uniformly asymptotically stable and the omega limit set contains no...

We study the scattering resonances between two confocal hyperbolae and show that the spectrum is dominated by the effect of a single periodic orbit. There are two distinct cases depending on whether the orbit is geometric or diffractive. A generalization of periodic orbit theory allows us to incorporate the second possibility. In both cases we also perform a WKB analysis. Although it is found t...

Let X be an Axiom A flow with a transverse torus T exhibiting a unique orbit O that does not intersect T . Suppose that there is no nullhomotopic closed curve in T contained in either the stable or unstable set of O. Then we show that X has either an attracting periodic orbit or a repelling periodic orbit or is transitive. In particular, an Anosov flow with a transverse torus is transitive if i...

An analytical approach to determine critical parameter values of homoclinic bifurcations in three-dimensional systems is reported. The homoclinic orbit is supposed to be a limit of a unique periodic orbit. Hence, the multiple scales perturbation method is performed to construct an approximation of the periodic solution and its frequency. Then, two simple criteria are used. The first criterion i...

Isochrons are foliations of phase space that extend the notion of phase of a stable periodic orbit to the basin of attraction of this periodic orbit. Each point in the basin of attraction lies on only one isochron, and two points on the same isochron converge to the periodic orbit with the same phase. Global isochrons, that is, isochrons extended into the full basin of attraction rather than ju...