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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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