Fractional Generalized Quasi-synchronization of Incommensurate Fractional Order Oscillators

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GS was introduced in Rulkov, Sushchik, Tsimring, Abarbanel (Phys Rev E 51, 980–994, 1995, [1]), but here definitions are extended and given in our own conception, for fractional order nonlinear systems, by using the fractional incommensurate differential primitive element. The problem of the Fractional Generalized Synchronization (FGS) was studied for a class of strictly different nonlinear commensurate fractional order systems in the master–slave configuration scheme (Martinez-Guerra, Mata-Machuca, Nonlinear Dyn Vol. 77, 1237–1244, 2014, [2]). Recently, numerous works have been reported on the problem of synchronization for incommensurate fractional order chaotic systems (Razminia, Majd, Baleanu, Adv Differ Equ 1–12, 2011, [3]), (Delshad, Asheghan, Beheshti, Commun Nonlinear Sci Numer Simul 3815–3824, 2011, [4]), (Wang, Zhang, Phys Lett A, 202–207, 2009, [5]), (Boulkroune, Bouzeriba, Bouden, Neurocomput Vol 173, 606–614, 2016, [6]). In general, study synchronization of strictly different systems is equivalent to study the asymptotic stability of the origin of the synchronization error or the stability of the synchronization manifold if possible. In many of these references, the stability of the incommensurate fractional order dynamics of the synchronization error is translated into a problem of stability of a commensurate fractional order or even an integer order system through a change of variable. In this chapter, we will show a convergence analysis directly from the incommensurate fractional order dynamics of the synchronization error. It is natural to present the incommensurate fractional order dynamics of the synchronization error in a modal decomposition due to each dynamics have different fractional order. Thus, we can obtain asymptotic convergence in a compact region near the origin in case of synchronization error for generalized synchronization of strictly different incommensurate fractional- order systems by using dynamical controllers obtained from differential algebraic techniques. In this chapter, the main contribution is a Fractional Generalized Synchronization constructive method for nonlinear incommensurate fractional-order chaotic systems in a master–slave topology, this phenomena is studied from an algebraic and differential point of view, that allows us to construct an Incommensurate Fractional Generalized Observability Canonical Form (IFGOCF) from an adequate selection of a fractional differential primitive element and moreover gives explicitly the form of the synchronization algebraic manifold for strictly different fractional order nonlinear systems. The former enables us to design an incommensurate fractional order dynamical controller able to achieve synchronization of strictly different incommensurate fractional order chaotic systems. Moreover, we introduce the concepts so-called Incommensurate Fractional Algebraic Observability and a fractional order Picard–Vessiot system. As far as we know, synchronization of strictly different incommensurate fractional order systems have not been tackled from this perspective.