Weak, modified and function projective synchronization of Cohen–Grossberg neural networks with mixed time-varying delays and parameter mismatch via matrix measure approach

  • Rakesh KumarEmail author
  • Subir Das
Original Article


This paper is concerned with the modified function projective synchronization of Cohen–Grossberg neural networks systems with parameter mismatch and mixed time-varying delays. Due to the existence of parameter mismatch between the drive and slave systems, complete modified function projective synchronization is not possible to achieve. So a new concept, viz., weak modified function projective synchronization, is discussed up to a small error bound. Several generic criteria are derived to show weak modified function projective synchronization between the systems. The estimation of error bound is done using matrix measure and Halanay inequality. Simulation results are proposed graphically for different particular cases to show the synchronization between parameter-mismatched systems, which validate the effectiveness of our proposed theoretical results.


Neural networks Mixed time-varying delays Halanay inequality Matrix measure Modified function projective synchronization 



The authors are extending their heartfelt thanks to the revered reviewers for their constructive suggestions toward up-gradation of the article.

Compliance with ethical standards

Conflict of interest

The authors Subir Das, Professor, Department of Mathematical Sciences, IIT (BHU), Varanasi, India, and Mr. Rakesh Kumar, who is pursuing his Ph.D. degree under the supervision of Prof. S. Das, declare that they have no conflict of interest.


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© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, Indian Institute of TechnologyBanaras Hindu UniversityVaranasiIndia

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