Page 656 - Mathematical Techniques of Fractional Order Systems

Fractional Order Chaotic Systems Chapter | 21 627

between two or more chaotic systems. The second is called master slave
or unidirectional coupling mode (Ouannas et al., 2017g). Its principle is to
choose a special configuration involving two coupled systems so that the
behavior of the second depends on the behavior of the first, but recipro-
cally, the first is not influenced by the behavior of the second. The first
system generating chaos is the emitter (master) system, and the second is
the receiver (slave) system. This is described by recurrent equations and
characterized by its state variables constituting the state vector.
Nevertheless, in spite of this multitude of research, the synthesis of the
fractional order chaotic systems remains an open problem in many fields of
research because of the nature of these systems which are, on the one hand,
considered as systems with long memory. On the other hand, they present a
complex dynamic. Thus, several experimental and theoretical studies show
that the problem of the synchronization of the fractional order systems in the
presence of chaos is still an open attractive subject to many scientists in
many fields of research.
In this chapter, the focus is meant to show that the recurrences associated
with fractional order chaotic systems are an efficient way of ensuring the
synchronization of continuous-time chaotic attractors. The efficiency for the
representation of the synchronization phenomenon of these systems will be
evaluated in terms of performance and robustness. The objectives are, first
of all, consisting of developing new synchronization schemes based on the
notion of recurrence for fractional order chaotic systems (Vladimirsky and
Ismailov, 2015a). Then, based on recurrence property (Afraimovich, 1999),
new topological synchronization criterions are derived between different
dimensional continuous-time in the case of fractional order chaotic systems
in different dimensions. Finally, several illustrative numerical applications
and computer simulations are used to confirm the theoretical results and to
prove the effectiveness of the proposed schemes.
The rest of this chapter is organized in the following way: a first sec-
tion, describes general definitions and preliminaries of nonlinear dynamic
systems, the theory of deterministic chaos, and the methods of Poincare ´
recurrence visualization and measures. The second section focuses on the
basic notions of fractional systems, namely the noninteger order derivation
in the Grunwald Letnikov sense, the fractional integral of
Riemann Liouville, the fractional derivation in Caputo’s sense, and the
stability condition of fractional order chaotic systems. The third section
exposes the topology of fractional order systems. The fourth section pre-
sents the problem of synchronization of fractional order chaotic systems
and their principal used methods. The fifth section illustrates the content of
the work, which consists of a new scheme on the topological synchroniza-
tion for some concrete examples on fractional order chaotic systems. At the
end, the work will be summarized by a general conclusion followed by a
bibliography.