Differential equations with pulses: existence and stability of periodic solutions

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.