Nonperiodic Trajectories Research Articles

Analysis of chaotic systems using the cell mapping approach

Chaotic motions in deterministic nonlinear systems are an important topic both from a theoretical and a practical point of view. In particular, there have been many studies of systems which yield bounded nonperiodic trajectories converging to attractors of a rather complicated nature, so-called strange attractors. Their existence was demonstrated in a class of nonlinear oscillators with periodic forcing which occur in electric circuit theory and mechanics. The determination of the domain of attraction of such attractors, depending on the parameters, is an interesting problem. It is shown, that the cell mapping approach, i.e., a discrete version of a Poincare map, represents a very efficient method for analyzing this problem.

Fourier transform method for the classical trajectory problem

By introducing an impluse potential at an external boundary, or by reflecting the scattering potential at such a boundary, collision dynamics is converted into an equivalent bound-state problem. Recent theoretical developments in classical bound-state dynamics. which make use of the geometrical structure in phase space known as an invariant toroid, can then be applied to collision problems. Fast Fourier transform techniques are used to construct the invariant toroid, which is the locus of an indefinitely extended nonperiodic classical trajectory. This approach to collision dynamics is applied to the collinear atom–oscillator problem modelling He–H2 vibrational excitation. Results are compared to previous classical and quantum dynamical studies. The proposed method obtains in a single calculation the complete ensemble of trajectories that correspond to a semiclassical wave function. At given total energy and for given global action integral values all trajectories are parallel straight lines when expressed in the coordinate space of true angle variables obtained by the Fourier transform calculation. These aspects of the formalism reduce the computation of inelastic transition probabilities and action integrals to simple geometric exercises.