We present and compare three generically applicable signal processing methods for periodic orbit quantization via harmonic inversion of semiclassical recurrence functions. In a first step of each method, a band-limited decimated periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. In a second step, the frequencies and amplitudes of the decimated signal ...

Ergodic Optimization is the process of finding invariant probability measures that maximize the integral of a given function. It has been conjectured that “most” functions are optimized by measures supported on a periodic orbit, and it has been proved in several separable spaces that an open and dense subset of functions is optimized by measures supported on a periodic orbit. All known positive...

We investigate the dynamics and transitions to extinction of hypercycles governed by periodic orbits. For a large enough number of hypercycle species (n>4) the existence of a stable periodic orbit has been previously described, showing an apparent coincidence of the vanishing of the periodic orbit with the value of the replication quality factor Q where two unstable (non-zero) equilibrium point...

Unstable periodic orbits occur naturally in many nonlinear dynamical systems. They can generally not be be observed directly, but a number of control schemes have been suggested to stabilize them. One such scheme is that by Pyragas [35,36,40], which uses time-delayed feedback to target a specific unstable periodic orbit of a given period and stabilize it. This paper considers the global effect ...

Indeed, in [3] they showed that, if one of the invariant circles of f is missing, then for some p and q with gcd(p; q) = 1 the map must have a periodic orbit of type (p; q) which is not a Birkhoff orbit. On the other hand, Boyland showed in [2] that a twistmap with a non Birkhoff periodic orbit of type (p; q) (gcd(p; q) = 1 must have positive topological entropy, which clearly implies the theorem.

We consider the billiard map inside a polyhedron. We give a condition for the stability of the periodic trajectories. We apply this result to the case of the tetrahedron. We deduce the existence of an open set of tetrahedra which have a periodic orbit of length four (generalization of Fagnano’s orbit for triangles), moreover we can study completely the orbit of points along this coding.

For a modified Kepler problem, we reexamine jumps in the saturation spectral rigidity and large oscillations of the level number variance with near zero minima. Earlier discrepancy between the periodic orbit theory and numerical calculation is cleared by a quantum mechanical calculation. A new class of radial periodic orbits is included establishing a complete correspondence between the periodi...

An infinite square well with a discontinuous step is one of the simplest systems to exhibit non-Newtonian ray-splitting periodic orbits. This system is analyzed using both time-independent perturbation theory (PT) and periodic-orbit theory and the approximate formulas for the energy eigenvalues derived from these two approaches are compared. The periodic orbits of the system can be divided into...