In this chapter, the basic concepts of fractional-order dynamical systems are presented, and the synchronization methodologies of fractional order chaotic dynamical systems are established using slide mode control techniques. Through observation of the different phase portraits and time-series graphs of fractional order finance systems through utilization of the fractional calculus and computer simulation, the authors have obtained that the lowest dimension of fractional order hyper chaotic finance system is 3.90, which is less than 4. Bifurcation diagrams and Lyapunov exponents of fractional order hyper chaotic finance system are calculated to justify the chaos in the systems. Synchronization of two identical fractional-order hyper chaotic finance systems are achieved using sliding mode control techniques.
TopIntroduction
In the last few decades, fractional calculus has established the potential tools to describe in many branches of sciences, engineering, and interdisciplinary fields. Many physical systems can be well defined by fractional-order differential equations which behave chaotically or hyper-chaotically, such as the fractional order Chen system (Li & Chen, 2004), the fractional order Lorenz system (Grigorenko Grigoren, 2003), the fractional-order Duffing system (Gao & Yu, 2005), the fractional order Rossler system (Li & Chen, 2004), the fractional order financial system (Wang, Huang Shi, 2011), the fractional-order hyperchaotic Chen system, the fractional-order hyperchaotic novel system (Matouk, 2009) and so on. The concept of fractional derivative is utilized to model the behavior of real systems in various fields of science and engineering including bioengineering, medicine, biological tissues, cardiac tissue, fluid mechanics, viscoelasticity, material science, etc (Wilkie, Drapaca & Sivaloganathan,2011; Podlubny, 2005; Jun-Guo, 2005; Chen, Liu, Ma & Zhang 2012; Das Gupta, 2012; Odibat, 2010; Kumar et al., 2019). The attributes of fractional order systems, for which they have gained popularity in the investigation of dynamical systems, is that they allow a greater degree of flexibility in the model. However, there are many differences between ordinary differential equation systems (integer-order) and corresponding fractional-order differential equation systems (Faieghi, Delavari & Baleanu, 2013).