Page 465 - Mathematical Techniques of Fractional Order Systems

Chapter 15

Multiswitching Synchronization

Between Chaotic Fractional
Order Systems of Different

Dimensions

2
1
Samuel T. Ogunjo , Kayode S. Ojo and Ibiyinka A. Fuwape 1,3
1 2
Federal University of Technology, Akure, Nigeria, University of Lagos, Lagos, Nigeria,
3
Michael and Cecilia Ibru University, Ughelli, Nigeria
15.1 INTRODUCTION

Chaos is defined as the aperiodic long-term behavior in a deterministic sys-
tem that exhibits sensitive dependence on initial conditions (Strogatz, 2000;
Azar et al., 2017b; Pham et al., 2017; Wang et al., 2017; Azar and
Vaidyanathan, 2015a,c, 2016). Chaotic systems are characterized by one or
more positive Lyapunov exponents. The study of chaotic systems have
evolved from the first discovered Lorenz system into maps, partial differen-
tial equations, delayed differential equations, fractional order systems, and
time series analyses (Ogunjo et al., 2017).
Due to the sensitivity to initial conditions, the existing belief before the
work of Pecora and Carroll (1990) was that the trajectory of two chaotic
systems cannot converge. Boccaletti et al. (2002) defined synchronization
of chaotic systems as a process wherein two (or many) chaotic systems
(either equivalent or nonequivalent) adjust a given property of their motion
to a common behavior due to a coupling or to a forcing. This implies that
given two chaotic systems (identical or nonidentical): the drive y i ðtÞ and
the response x i ðtÞ. The trajectory of the two systems will differ due to
sensitivity to initial condition. However, if suitable control u i ðtÞ is added to
x i ðtÞ, the trajectory of x i ðtÞ canbemadetocoincidewiththatof y i ðtÞ
such that
lim jjy i ðtÞ 2 xÞiðtÞjj 5 0; ’t $ 0
t-N

Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00015-5
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