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International Journal of Differential Equations
Volume 2017, Article ID 1946304, 7 pages
https://doi.org/10.1155/2017/1946304
Research Article

Finite Time Synchronization of Extended Nonlinear Dynamical Systems Using Local Coupling

1School of Mathematical Sciences and Information Technology, Department of Mathematics, Yachay Tech, Urcuqui, Ecuador
2Facultad de Ingeniería en Ciencias Aplicadas, Universidad Técnica del Norte, Ibarra, Ecuador
3Departamento de Estadística, Facultad de Ciencias Económicas y Sociales, Universidad Central de Venezuela, Caracas, Venezuela

Correspondence should be addressed to P. García; ce.ude.ntu@aicragp

Received 21 July 2017; Accepted 14 November 2017; Published 7 December 2017

Academic Editor: Peiguang Wang

Copyright © 2017 A. Acosta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider two reaction-diffusion equations connected by one-directional coupling function and study the synchronization problem in the case where the coupling function affects the driven system in some specific regions. We derive conditions that ensure that the evolution of the driven system closely tracks the evolution of the driver system at least for a finite time. The framework built to achieve our results is based on the study of an abstract ordinary differential equation in a suitable Hilbert space. As a specific application we consider the Gray-Scott equations and perform numerical simulations that are consistent with our main theoretical results.