Differential equations with pulses: existence and stability of periodic solutions

TL;DR

This paper develops an algorithm to find and analyze the stability of periodic solutions in impulsive differential equations with periodic pulses, providing a comprehensive understanding of their long-term behavior.

Contribution

The authors introduce a novel method to locate and study the stability of periodic solutions in impulsive systems with periodic pulses, including bifurcation analysis.

Findings

All periodic solutions can be explicitly characterized using the time-$$ map.

The stability of solutions can be determined systematically.

The approach allows full characterization of the system's asymptotic dynamics.

Abstract

We consider generic differential equations in $\mathbb{R}$ with a finite number of hyperbolic equilibria, which are subject to $\omega$--periodic instantaneous perturbative pulses ($\omega>0$). Using the time-$ \omega$ map of the original system (without perturbation), we are able to find all periodic solutions of the perturbed system and study their stability. In this article, we establish an algorithm to locate $\omega$--periodic solutions of impulsive systems of frequency $\omega$, to study their stability and to locate Saddle-node bifurcations. With our technique, we are able to fully characterise the asymptotic dynamics of the system under consideration.