Topological Conjugacy and Structural Stability in Hybrid Rational-Exponential Dynamical Systems

This paper investigates the concepts of topological conjugacy and structural stability within the framework of hybrid dynamical systems that incorporate both rational and exponential functions. These systems, characterized by a combination of continuous flows governed by differential equations and discrete jumps, model a wide range of phenomena but present significant analytical challenges. The primary objective is to establish a classification of these systems based on topological conjugacy, which provides an equivalence relation preserving the qualitative structure of their dynamics. We define a class of hybrid rational-exponential maps and develop criteria under which two such maps are topologically conjugate. Furthermore, we analyze the structural stability of these systems, defined as the persistence of their topological structure under small perturbations of the governing functions. The methodology involves constructing a suitable function space for the hybrid maps and employing techniques from perturbation theory and complex analysis. The main results provide sufficient conditions for both the existence of a topological conjugacy and for a system to be structurally stable. These findings are discussed in the context of classical results for purely rational or purely exponential dynamics, highlighting the unique behaviors that arise from the hybrid interaction. This work contributes to a deeper theoretical understanding of complex hybrid systems and lays the groundwork for analyzing their behavior in practical applications.