Given an object in a category, we study its orbit algebra with respect to an endofunctor. We show that if the object is periodic, then its orbit algebra modulo nilpotence is a polynomial ring in one variable. This specializes to a result on Ext-algebras of periodic modules over Gorenstein algebras. We also obtain a criterion for an algebra to be of wild representation

Classical and semiclassical periodic orbit expansions are applied to the dynamics of a point particle scattering elastically oo several disks in a plane. Fredholm determinants, zeta functions, and convergence of their cycle expansions are tested and applied to evaluation of classical escape rates and quantum resonances. The results demonstrate applicability of the Ruelle and Gutzwiller type per...

Classical chaotic systems with symbolic dynamics but strong pruning present a particular challenge for the application of semiclassical quantization methods. In the present study we show that the technique of periodic orbit quantization by harmonic inversion of trace formulae, which does not rely on the existence of a complete symbolic dynamics or other specific properties, lends itself ideally...

The Lindstedt–Poincaré technique in perturbation theory is used to calculate periodic orbits of perturbed differential equations. It uses a nearby periodic orbit of the unperturbed differential equation as the first approximation. We derive a numerical algorithm based upon this technique for computing periodic orbits of dynamical systems. The algorithm, unlike the Lindstedt–Poincaré technique, ...

We demonstrate with a minimal example that in Filippov systems (dynamical systemsgoverned by discontinuous but piecewise smooth vector fields) stable periodic motionwith sliding is not robust with respect to stable singular perturbations. We considera simple dynamical system that we assume to be a quasi-static approximationof a higher-dimensional system containing a fast sta...

We consider an autonomous differential system in Rn with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of n − 1 codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of t...

The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of the unstable periodic orbits embedded in that set. This result has been proven to be valid for hyperbolic chaotic systems. We test the goodness of such a periodic-orbit characterization of the natural measure for nonhyperbolic chaotic systems by comparing the natural measure of a typical c...

We consider an autonomous differential system in Rn with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of n − 1 codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of t...

The well-established method of phase reduction neglects information about a limit-cycle oscillator's approach towards its periodic orbit. Consequently, phase reduction suffers in practicality unless the magnitude of the Floquet multipliers of the underlying limit cycle are small in magnitude. By defining isostable coordinates of a periodic orbit, we present an augmentation to classical phase re...