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Chapter 12





             Dual Combination


             Synchronization Scheme for
             Nonidentical Different


             Dimensional Fractional Order
             Systems Using Scaling Matrices




             Vijay K. Yadav, Mayank Srivastava and Subir Das
             Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, India


             12.1 INTRODUCTION
             The fractional calculus is a name for the theory of integrals and derivatives
             of arbitrary real order and also of complex order, which unify and generalize
             the notions of differintegral orders. Fractional calculus can be used in many
             modeling and design problems. A dynamical system involving fractional
             order time derivatives is known as a fractional dynamical system.
             Introduction of fractional calculus in nonlinear models has rendered a new
             dimension to the existing problems. Again, due to the nonlocal property of a
             fractional order differential operator, it takes into account the fact that the
             future state depends upon the present state as well as all of the history of its
             previous states. For this realistic property, the fractional order systems are
             becoming popular. Another reason behind using fractional order derivatives
             is that these are naturally related to the systems with memory which prevail
             for most of the physical and scientific system models. The fractional deriva-
             tive of a function depends on the values of the function over the entire inter-
             val. Thus, it is suitable for modeling of the systems with long-range
             interactions both in space and time. Fractional derivative has the flexibility
             to allow incorporation of different types of information. The fractional calcu-
             lus which was in the earlier stage considered as a mathematical curiosity
             now becomes the object for the extensive development of fractional order
             partial differential equations for its applications in various physical areas
             of sciences and engineering. Geometric and physical interpretations of



             Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00012-X
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