2.1.1 Several Important Functions of Fractional Calculus

In this subsection, three important functions of fractional calculus will be introduced: the gamma function, the beta function, and the Mittag–Leffler (ML) function.

On the basis of Equation (2.2), we have

2.3

2.1.2 Fractional Integral and Derivatives

In this subsection, we will introduce the definitions of the fractional integral and the three main fractional derivatives, i.e., the Grünwald–Letnikov (GL), Riemann–Liouville (RL), and Caputo definitions [154] [163–165].

In this definition, the initial time is set to zero. The same is true for the following definitions.

In this book, we use the definition of the Caputo fractional derivative to analyze the problems of control and synchronization control for fractional-order systems. Furthermore, we will use instead of .

2.1.3 Some Important Lemmas

In this subsection, some lemmas are introduced, which will be used to analyze the control for fractional-order systems in the book.

2.2.1 Fractional-Order Lorenz System

From the simplified equation of convective rolls in the equations of the atmosphere, the first three-dimensional chaotic system was derived by Lorenz in 1963 [178]. Furthermore, the developed fractional-order Lorenz system [179] is given as follows:

2.46

where , , and are fractional orders, , , and are system state variables, and , , and are system parameters. For fractional orders chosen as , system parameters given by , , and , and initial conditions chosen as , the simulation results of the fractional-order Lorenz system are shown in Figure 2.1. Furthermore, the bifurcation diagram for is presented in Figure 2.2. Following the bifurcation diagram (Figure 2.2), the dynamic characteristic of two cycles for is shown in Figure 2.3.

2.2.2 Fractional-Order Van Der Pol Oscillator

The van der Pol oscillator is a non-conservative oscillator with nonlinear damping. The fractional-order van der Pol oscillator is given in the following form [164]:

2.47

where and are fractional orders, and are system state variables, and is a system parameter. For fractional orders chosen as and , the system parameter given by , and initial conditions chosen as and , the simulation results of the fractional-order van der Pol oscillator are given in Figure 2.4.

2.2.3 Fractional-Order Genesio–Tesi System

The Genesio-Tesi system was first described using mathematical equations by Petrás̆ [164]. In addition, the fractional-order Genesio–Tesi system is defined as follows [180]:

2.48

where , , and are fractional orders, , , and are system state variables, and , , and are system parameters. For fractional orders chosen as and , system parameters given by , , and , and initial conditions chosen as , , and , the simulation results of the fractional-order Genesio–Tesi system are presented in Figure 2.5. Furthermore, the bifurcation diagram for is presented in Figure 2.6, with the initial conditions , , and . Following the bifurcation diagram (Figure 2.6), the dynamic characteristic of two cycles for is shown in Figure 2.7.

2.2.4 Fractional-Order Arneodo System

The fractional-order Arneodo system is described as follows [181]:

2.49

where , , and are fractional orders, , , and are system state variables, and , , and are system parameters. For fractional orders chosen as , , and , system parameters given by , , and , and initial conditions chosen as , , and , the simulation results of the fractional-order Arneodo system are shown in Figure 2.8. Furthermore, the bifurcation diagram for is presented in Figure 2.9. Following the bifurcation diagram (Figure 2.9), the dynamic characteristic of one cycle for is shown in Figure 2.10.

2.2.5 Fractional-Order Lotka–Volterra System

A two-predator and one-prey generalization of the Lotka–Volterra system was proposed by Samardzija and Greller [182]. Its fractional-order model is given as follows [164]:

2.50

where , , and are fractional orders, , , and are system state variables, and , , and are system parameters. For fractional orders chosen as , system parameters given by , , and and initial conditions chosen as , , and , the simulation results of the fractional-order Lotka–Volterra system are presented in Figure 2.11. Furthermore, the bifurcation diagram for is presented in Figure 2.12. Following the bifurcation diagram (Figure 2.12), the chaotic dynamic characteristic for is shown in Figure 2.13.

2.2.6 Fractional-Order Financial System

Ma and Chen [183] gave a simplified finance model. According to the integer-order finance model, the fractional-order financial system is described as follows [8]:

2.51

where , , and are fractional orders, , , and are system state variables, and , , and are system parameters. For fractional orders chosen as , system parameters given by and , and initial conditions chosen as , , and , the simulation results of the fractional-order financial system are given in Figure 2.14. Furthermore, the bifurcation diagram for is presented in Figure 2.15. Following the bifurcation diagram (Figure 2.15), the dynamic characteristic of one cycle for is shown in Figure 2.16.

2.2.7 Fractional-Order Newton–Leipnik System

The Newton–Leipnik system is described by a nonlinear differential equation in [164]. By considering the fractional calculus, the fractional-order Newton-Leipnik system is given by Sheu et al. [184] as follows:

2.52

where , , and are fractional orders, , , and are system state variables, and and are system parameters. For fractional orders chosen as , system parameters given by and , and initial conditions chosen as , , and , the simulation results of the fractional-order Newton–Leipnik system are shown in Figure 2.17. Furthermore, the bifurcation diagram for is presented in Figure 2.18. Following the bifurcation diagram (Figure 2.18), the chaotic dynamic characteristic for is shown in Figure 2.19.

2.2.8 Fractional-Order Duffing System

The fractional-order Duffing system is written as follows [164]:

2.53

where and are fractional orders, and are system state variables, and , , and are system parameters. For fractional orders chosen as and , system parameters given by , , and , and initial conditions chosen as and , the simulation results of the fractional-order Duffing system are given in Figure 2.20. Furthermore, the bifurcation diagram for is presented in Figure 2.21. Following the bifurcation diagram (Figure 2.21), the dynamic characteristic of one cycle for is shown in Figure 2.22.

2.2.9 Fractional-Order Lü System

The Lü system [185] is known as a bridge between the Lorenz system [178] and the Chen system [186]. Its fractional-order differential equation is described as follows [187]:

2.54

where , , and are fractional orders, , , and are system state variables, and , , and are system parameters. For fractional orders chosen as , system parameters given by , , and , and initial conditions chosen as , , and , the simulation results of the fractional-order Lü system are presented in Figure 2.23. Furthermore, the bifurcation diagram for is presented in Figure 2.24. Following the bifurcation diagram (Figure 2.24), the dynamic characteristic of one cycle for is shown in Figure 2.25.

2.2.10 Fractional-Order Three-Dimensional System

On the basis of the fractional-order Lorenz system [179], a three-dimensional fractional-order system is written as follows [188]:

2.55

where , , and are fractional orders, , , and are system state variables, and , , and are system parameters. For fractional orders chosen as , , and , system parameters given by , , and , and initial conditions chosen as , , and , the simulation results of the fractional-order three-dimensional system are shown in Figure 2.26. Furthermore, the bifurcation diagram for is presented in Figure 2.27. Following the bifurcation diagram (Figure 2.27), the dynamic characteristic of one cycle for is shown in Figure 2.28.

2.2.11 Fractional-Order Hyperchaotic Oscillator

The fractional-order hyperchaotic oscillator is described by the following form [189]:

2.56

where , , , , and are fractional orders, , , , and are system state variables, and , , , , , and are system parameters. For fractional orders chosen as , system parameters given by , , , , , and , and initial conditions chosen as , , , and , the simulation results of the fractional-order hyperchaotic oscillator are given in Figure 2.29. Furthermore, the bifurcation diagram for is presented in Figure 2.30. According to the bifurcation diagram (Figure 2.30), the dynamic characteristic of one cycle for is shown in Figure 2.31.

2.2.12 Fractional-Order Four-Dimensional Hyperchaotic System

The fractional version of the four-dimensional hyperchaotic system is given by the following [190]:

2.57

where , , , and are fractional orders, , , , and are system state variables, and , , , , and are system parameters. For fractional orders chosen as , system parameters given by , , , , and , and initial conditions chosen as , , , and , the simulation results of the fractional-order four-dimensional hyperchaotic system are presented in Figure 2.32. Furthermore, the bifurcation diagram for is presented in Figure 2.33. Following the bifurcation diagram (Figure 2.33), the dynamic characteristic of one cycle for is shown in Figure 2.34.

2.2.13 Fractional-Order Hyperchaotic Cellular Neural Network

The model of a fractional-order hyperchaotic cellular neural network can be written as follows [191]:

2.58

where , , , and are fractional orders, , , , and are system state variables, and , , and are system parameters. For fractional orders chosen as , system parameters given by , , and , and initial conditions chosen as , , , and , the simulation results of a fractional-order hyperchaotic cellular neural network are shown in Figure 2.35. Furthermore, the bifurcation diagram for is presented in Figure 2.36. Following the bifurcation diagram (Figure 2.36), the chaotic dynamic characteristic for is shown in Figure 2.37.