By Rafael Martínez-Guerra, Claudia A. Pérez-Pinacho, Gian Carlo Gómez-Cortés

This ebook offers a basic assessment of numerous techniques of synchronization and brings jointly similar ways to safe communique in chaotic structures. this is often accomplished utilizing a mix of analytic, algebraic, geometrical and asymptotical easy methods to take on the dynamical suggestions stabilization challenge. particularly, differential-geometric and algebraic differential thoughts show very important structural houses of chaotic structures and function advisor for the development of layout approaches for a wide selection of chaotic structures. the fundamental differential algebraic and geometric recommendations are offered within the first few chapters in a singular means as layout instruments, including chosen experimental reports demonstrating their value. the next chapters deal with fresh purposes. Written for graduate scholars in utilized actual sciences, platforms engineers, and utilized mathematicians attracted to synchronization of chaotic platforms and in safe communications, this self-contained textual content calls for simply simple wisdom of integer usual and fractional traditional differential equations. layout purposes are illustrated with assistance from numerous actual types of useful interest.

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Circ. Syst. I Reg. Pap. 55(6), 1685–1694 (2008) 12. P. Gaspard, in Encyclopedia of Nonlinear Science, ed. by A. Scott (Routledge, New York, 2005), pp. 808–811 References 31 13. A. Isidori, Nonlinear Control Systems, 3rd edn. Communications and Control Engineering (Springer, New York, 1995) 14. M. Javidi, N. Nyamoradi, Numerical chaotic behaviour of the fractional Rikitake system. World Acad. Union 9, 120–129 (2013) 15. K. Khalil, Nonlinear Systems (Prentice Hall, Upper Saddle River, 2002), pp. 4803–4811 16.

Gegovska-Zajkova, S. Kostadinova, On Chua Dynamical System, vol. 2 (Scientific Publications of the State University of Novi Pazar, Novi Pazar, 2010), pp. 53–60 17. R. Kolchin, Differential Algebra and Algebraic Groups (Academic, New York, 1973) 18. R. L. Mata-Machuca, Generalized synchronization via the differential primitive element. Appl. Math. Comput. 232, 848–857 (2014) 19. L. Mata Machuca, R. Martínez Guerra, R. López Aguilar, Observadores para Sincronización de Sistemas Caóticos: Un Enfoque Diferencial y Algebraico (Editorial Académica Española, Saarbrücken, 2013) 20.

26 1 Control Theory and Synchronization Fig. 10 Why Fractional Order? The question then is this: can we give a deterministic natural assumption in some known cases to justify the use of fractional systems? The answer to this question is yes. 41) where k1 , k2 , 1 , 2 , ˛1 y ˛2 are positives constants. In this model, we assume the following two hypotheses: 1. t/ D C2 e k2 t . Then the evolution of the populations over time follows a classical exponential law. 2. When one population grows beyond a certain point, then the other decreases, and conversely, which justifies the nonlinear terms of type x.