Homoclinic and Heteroclinic Trajectories of Differential Equations with Piecewise Constant Arguments of Generalized Type
Mehmet Onur Fen, Fatma Tokmak Fen
TL;DR
The paper addresses EPCAG, a class of differential equations with discontinuous right-hand sides governed by a discrete-time map, and proves the existence of homoclinic and heteroclinic trajectories in a functional sense. It employs a Banach fixed point framework to construct unique, uniformly bounded solutions $\phi_\alpha$ for each bounded discrete orbit $\alpha$, and develops stable/unstable sets $W^s$ and $W^u$ to transfer homoclinic/heteroclinic structure from the discrete map to the continuous dynamics. A key contribution is establishing that, under suitable contraction conditions, homoclinic/heteroclinic relations in the discrete map imply corresponding relations among the bounded solutions, with hyperbolicity carried over to the solution set when the discrete map is hyperbolic. An illustrative example based on the logistic map demonstrates homoclinic and heteroclinic motions within EPCAG, validating the theoretical framework and highlighting potential for chaotic-like behavior in hybrid systems. The work broadens the understanding of long-term dynamics in systems with piecewise constant arguments and sets the stage for extensions to PDEs and other hybrid models.
Abstract
Quasilinear systems with piecewise constant arguments of generalized type are under investigation from the asymptotic point of view. The systems have discontinuous right-hand sides which are identified via a discrete-time map. It is rigorously proved that homoclinic and heteroclinic solutions are generated, and they are taken into account in the functional sense. The Banach fixed point theorem is used for the verification. The hyperbolic set of solutions is also discussed, and an example supporting the theoretical findings is provided.