The aim of this paper is to study the long-term behavior of a class of self-interacting diffusion processes on ℝd. These are solutions to SDEs with a drift term depending on the actual position of the process and its normalized occupation measure μt. These processes have so far been studied on compact spaces by Benaïm, Ledoux and Raimond, using stochastic approximation methods. We extend these methods to ℝd, assuming a confinement potential satisfying some conditions. These hypotheses on the confinement...

We consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 {ζ→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits - the shadowing chain lemma - via minimization of action integrals and using simple geometrical arguments.

This article presents a theorem about the topological conjugacy of a gradient dynamical system with a constant time step and the cascade generated by its Euler method. It is shown that on the two-dimensional sphere S² the gradient dynamical flow is, under some natural assumptions, correctly reproduced by the Euler method for a sufficiently small time step. This means that the time-map of the induced dynamical system is globally topologically conjugate to the discrete dynamical system obtained via...

We prove that for every nonempty compact manifold of nonzero dimension no self-homeomorphism and no continuous self-mapping has the uniform pseudo-orbit tracing property. Several relevant counterexamples for recently studied hypotheses are indicated.