We extend the application of the concept of structural invariance to bounded timeindependent systems. This concept, previously introduced by two of us to argue that the connection between random matrix theory and quantum systems with a chaotic classical counterpart is in fact largely exact in the semiclassical limit is extended to the energy spectra of bounded timeindependent systems. We procee...

The reduction of smooth dynamical systems to lower dimensional center manifolds containing the essential bifurcation dynamics is a very useful approach both for theoretical investigations as well as for numerical computation. Since this approach relies on smoothness properties of the system and on the existence of a basic linearization the question arises if this approach can be carried over to...

We show that, for a certain class of systems, the problem of establishing the existence of periodic orbits can be successfully studied by means of a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate positions of periodic points, and the existence of periodic orbits in a neighborhood of these approximations is proved using an interval operat...

We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problemdependent critical value so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric period...

We consider a generalized Gause prey-predator model with T -periodic continuous coefficients. In the case where the Poincaré map P over time T is well defined, the result of the paper can be explained as follows: we locate a subset U of R2 such that the topological degree d(I −P,U) equals to +1. The novelty of the paper is that the later is done under only continuity and (some) monotonicity ass...

Abstract. We analyze the dynamics of a deterministic model of inhibitory neuronal networks proving that the discontinuities of the Poincaré map produce a never empty chaotic set, while its continuity pieces produce stable orbits. We classify the systems in three types: the almost everywhere (a.e.) chaotic, the a.e. stable, and the combined systems. The a.e. stable are periodic and chaos appears...

The dynamics of a hysteretic relay oscillator with simple harmonic forcing is studied in this paper. Even though there are no bounded solutions in the absence of forcing, periodic excitation gives rise to more complex responses including periodic, quasi-periodic, and chaotic behavior. A Poincaré map is introduced to facilitate mathematical analysis. Families of period-one solutions are determin...

Abstract We introduce the notion of a strong pair periodic segments over [0, T ] and we show its applications in detecting chaotic dynamics. prove various number fixed point index formulas concerning points Poincaré map.