Impulses in Differential Equations and Dynamical Systems

Many systems present a dynamic behavior that is characterized by the fact that at certain times they undergo a sudden change throughout their evolution. This situation is going to be studied in this memory. On the one hand, different techniques will be used to study some boundary value problems for impulsive differential equations. This type of differential equations presents new and unexpected behaviors even in simple cases. On the other hand, the asymptotic behavior and attractors for dynamical systems with impulses will also be studied, mainly the case of evolution processes.2023-01-1

Pulse positive periodic solutions for some classes of singular nonlinearities

For given nonlinear differential equations it may occur that there are no periodic solutions. By introducing impulses at prescribed instants, periodic solutions may appear. We consider some impulsive nonlinear first order differential equations having periodic solutions and extend previous results to a larger class of nonlinearities.Ministerio de Economía, Industria y CompetitividadDepto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes

In this paper, following the work done in [11], we deal with the upper and weak-lower semicontinuity of pullback attractors for impulsive evolution processes. We first deal with the upper semicontinuity, presenting the abstract theory and applying it to uniform perturbations of a nonautonomous integrate-and-fire neuron model. We also present the abstract theory of weak-lower semicontinuity, and finish with an improvement of [11,Subsection 4.2], proving an invariance property for impulsive pullback omega-limits with weaker assumptions.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems

This is the accepted version of the following article: Everaldo de Mello Bonotto, Matheus Cheque Bortolan, Rodolfo Collegari, José Manuel Uzal. Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems. Discrete and Continuous Dynamical Systems - B, 2021, 26(9): 4645-4661. doi: 10.3934/dcdsb.2020306In this paper we investigate the long time behavior of a nonautonomous dynamical system (cocycle) when its driving semigroup is subjected to impulses. We provide conditions to ensure the existence of global attractors for the associated impulsive skew-product semigroups, uniform attractors for the coupled impulsive cocycle and pullback attractors for the associated evolution processes. Finally, we illustrate the theory with an application to cascade systems.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu