Page 467 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 15 453

(Golmankhaneh et al., 2015). Different concepts such as population, anoma-
lous diffusion, lithium ion battery model, viscoelasticity, and control systems
have been found to be better described by fractional calculus (Rivero et al.,
2011; Jiang et al., 2017; Azar et al., 2018a; Meghni et al., 2017a,b,c;
Tolba et al., 2017; Soliman et al., 2017).
Several studies have been conducted on the synchronization of fractional
order systems, multiswitching synchronization, and different order systems
(Azar et al., 2017a, 2018b; Singh et al., 2017; Ouannas et al., 2016, 2017a,b,c,d,
e,f,g,h; Grassi et al., 2017; Ouannas et al., 2017i). This current work aims to
implement multiswitching synchronization of two fractional order systems with
different dimensions. Related works to this research are stated in Section 15.2,a
description of the chaotic systems used is made in Section 15.3, the design of
controllers will be carried out in Section 15.4, the results obtained are explained
in Section 15.5, and conclusions are drawn in Section 15.6.

15.2 RELATED WORK

The synchronization of chaotic systems with different dimensions has been
studied for integer order systems. Some of the studies include increased order
synchronization of two systems (Ogunjo, 2013; Ojo et al., 2014c), hybrid
function projective combination synchronization (Ojo et al., 2014b), reduced
order function projective combination synchronization (Ojo et al., 2014a),
backstepping fuzzy adaptive control (Wang and Fan, 2015), and experimental
designs (Adelakun et al., 2017)(Fig. 15.1).
Fractional order synchronization has been carried out using adaptive con-
trol (Hajipour and Aminabadi, 2016), generalized synchronization (Lu, 2005;
Ge and Ou, 2008), multiswitching synchronization (Vincent et al., 2015),
active sliding mode controller (Tavazoei and Haeri, 2008), nonidentical cha-
otic fractional order systems (Golmankhaneh et al., 2015), inverse matrix
projective synchronization Ouannas et al. (2017c), and a new matrix scaling
method for multidimensional synchronization.

15.3 SYSTEM DESCRIPTION
15.3.1 Duffing Oscillator

One of the most widely studied dynamical systems is the Duffing oscillator.
It is given as

: dVðyÞ
€
y1 by 1 5 gðf; ω; tÞ ð15:4Þ
dy
where the variable y is the displacement from the equilibrium position, f, b,
g, and ω are the forcing strength, damping parameters, periodic driving force,
and the angular frequency, respectively Ogunjo (2013).