In this paper, we prove that for every Finsler n-sphere (S, F ) for n ≥ 3 with reversibility λ and flag curvature K satisfying ( λ λ+1 )2 < K ≤ 1, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp(πiμ) with an irrational μ. Furthermore, there always exist three...

We investigate the relation between the chaotic dynamics and the hierarchical phase-space structure of the standard map as an example for generic Hamiltonian systems with a mixed phase space. We demonstrate that even in ideal situations when the phase-space structure is dominated by a single scaling, the long-time dynamics is not dominated by this scaling. This has consequences for the power-la...

In this paper, chaotic behavior study of a five-cells chopper when it is associated to a nonlinear load is reported. The model of such system is described by a piece-wise smooth six dimensional non-autonomous system. Then, some basic dynamical properties, such as Poincaré mapping, first return map and continuous spectrum are investigated to highlight the main characteristic of this particular c...

The Hodgkin–Huxley model was developed to characterize the action potential of a squid axon. It has served as an archetype for compartmental models of the electrophysiology of biological membranes. Thus the dynamics of the Hodgkin–Huxley model have been extensively studied both with a view to their biological implications and as a test bed for numerical methods that can be applied to more compl...

We prove that return time statistics of a dynamical system do not change if one passes to an induced (i.e. rst return) map. We apply this to show exponential return time statistics in i) smooth interval maps with nowhere{dense critical orbits and ii) certain interval maps with neutral xed points. The method also applies to iii) certain quadratic maps of the complex plane.

The behaviour of light rays obeying Snell’s Law in a medium made up of two materials with different refractive indices and which are arranged in a periodic chessboard pattern is described. The analysis is in some ways analogous to the study of rational billiards and uses a return map on one surface to prove, amongst other things, that the number of angles with which any individual ray intersect...

We construct an obstruction theory for relative Hilbert schemes in the sense of [BF] and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface V , our obstruction theory determines a virtual fundamental class [[Hilb m V ]] ∈ A m(m−k) 2 (Hilb m V), which we use to define Poincaré invariants These maps are invaria...

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a...

In this paper, a novel West Nile Virus model looking upon the infected birds as monitoring threshold, for the mosquitoes and birds with impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control via comprehensively qualitative analysis. By using the Poincaré map, we obtain that the s...

Wave packets in a system governed by a Hamiltonian with a generic nonlinear spectrum typically exhibit both full and fractional revivals. It is shown that the latter can be eliminated by inducing suitable geometric phases in the states, by varying the parameters in the Hamiltonian cyclically with a period T . Further, with the introduction of this natural time step T , the occurrence of near re...