Page 427 - Mathematical Techniques of Fractional Order Systems

412 Mathematical Techniques of Fractional Order Systems

operator are Grunwald Letnikov (GL), Riemann Liouville (RL), and
Caputo which are defined as (Kilbas et al., 2006):

t2a
h ½Š
P k α
ð21Þ fðt 2 khÞ
α k50 k
D fðtÞ 5 lim ; t 2 a 5 nh; ð14:2aÞ
a t α
h-0 h
α d m 1 ð t m2α21
D fðtÞ 5 ðt2τÞ fðτÞdτ ; ð14:2bÞ
a t m
dt Γðm 2 αÞ a
α
C D fðtÞ 5 1 ð t m2α21 ðmÞ
a t ðt2τÞ f ðτÞdτ; ð14:2cÞ
Γðm 2 αÞ a
1
respectively, where αAR and m 2 1 # α # m.If α , 0 then the GL defini-
tion represents a fractional order integral. Applications of fractional calculus
are not limited to: bioengineering (Yousri et al., 2017), control (Haji and
Monje, 2017), analog filters (Said et al., 2016a; AbdelAty et al., 2017;
Radwan and Fouda, 2013; Soltan et al., 2015, 2014), oscillators (Said et al.,
2015a,b, 2016b, 2017, 2014), circuit theory (Radwan and Salama, 2011,
2012; Radwan, 2013; AbdelAty et al., 2016, 2018; Rashad et al., 2017;
Radwan et al., 2017a), chemistry (Mizrak and Ozalp, 2017), and image pro-
cessing (Raghunandan et al., 2017).
Inspired by this new rediscovered fractional order models concept,
researchers have proposed either fractional variations of integer order chaotic
system or new fractional order systems such as: fractional order Chen system
(Asheghan et al., 2011; Wang et al., 2006), fractional order Lorenz system
(Grigorenko and Grigorenko, 2003; Xi et al., 2014), FO Rosslers system (Li
and Chen, 2004), Coullet system (Shahiri et al., 2012), Liu system
(Daftardar-Gejji and Bhalekar, 2010), modied Van der Pol Dufng system
(Barbosa et al., 2007), and others (He and Chen, 2017a; Radwan et al., 2014;
Tolba et al., 2017; He and Chen, 2017b). However, there are new criteria
introduced by Sprott (2011) that any new chaotic system to be proposed
should achieve one of the following (Borah and Roy, 2017):
1. The system has to be a model for an important under investigated physi-
cal system and give some insight into its dynamics.
2. The system should have unprecedented dynamical behavior.
3. When compared with older systems describing the same dynamical phe-
nomena, the new one should be simpler.
This chapter is an excursion into fractional order continuous-time
chaotic systems and their engineering applications. The rest of this chapter
is organized as follows: Section 14.2 presents the two common simulations
schemes used in literature to solve fractional order differential equations
(FDEs); Sections 14.3 and 14.4 discusses selected applications in control