Synchronization Problem of a Novel Fractal-Fractional Orders’ Hyperchaotic Finance System

This paper investigates the synchronization problem of a novel fractal-fractional (FF) orders’ hyperchaotic ﬁnance system with model uncertainty and external disturbance. Firstly, a controller is designed to realize the synchronization of the nominal FF-orders’ hyperchaotic ﬁnance system. Secondly, a suitable ﬁler is designed to estimate uncertainty and disturbance, and then, the uncertainty and disturbance estimator- (UDE-) based control method is proposed to realize the synchronization problem of such system. Finally, numerical simulations are carried out to verify the correctness and the eﬀectiveness of the obtained results.


Introduction
e fractional calculus was introduced in 1695, and it is the generalization of integer-order calculus. Fractional calculus is a research hotspot in many scientific fields, especially in mathematics and engineering. Different to the integer-order calculus, fractional derivatives can describe long-term memory, as detailed in [1][2][3][4][5][6]. Chaotic motion is an advanced form of complex motion. Its most important characteristic is its high sensitivity to initial values, that is, small differences in initial values will lead to huge differences in system states. Since Lorenz proposed the first chaotic system in 1963, many researchers have begun to study this chaotic phenomenon. Over the past few decades, chaos and fractals have been treated differently for different purposes. Chaotic theory was introduced to capture multifaceted systems that exhibit impulsive randomness and are very sensitive to small changes in conditions. Fractals are created to replicate infinitely complex patterns that are self-similar at different scales. In recent years, the FF-orders' problem has been expressed in [7][8][9][10][11][12][13][14][15][16][17][18].
e results show that the FF-order model is more suitable for practical problems than the integer-order model. In recent years, the synchronization of fractional-order chaotic systems has attracted great attention, and various control methods have been proposed, such as adaptive control [19,20], active control [21], passive control [22], and sliding model control [23]. Although scholars have made great efforts in the control of fractionalorder chaotic systems, there are still many challenges and problems to be solved. For example, the uncertainty of the system has not taken into account the control channels and control technologies designed in many controllers and control combinations [24][25][26]. It is well known that chaotic systems are very sensitive to parametric and external perturbations. erefore, it is difficult to synchronize chaotic systems with parametric perturbations and external perturbations. Fortunately, some work has been done on the synchronization problem of integer-order chaotic systems with parametric and external perturbations. But, the results of synchronization research for chaotic systems with model uncertainty and external disturbance have some limitations, such as model uncertainties and external perturbations are assumed to be bounded, and these bounds are usually small. Moreover, the obtained method is based on linear matrix inequality (LMI) tools, thus the obtained results are conservative in some sense. Recently, the UDE-based control method has shown some advantages over the aforementioned results, see [27][28][29][30][31][32][33][34]. erefore, we shall apply the existing UDE-based control method to study the synchronization problem of the FF hyperchaotic finance system. Inspired by the above discussion, we consider the newly defined FF-operators of fractional calculus, which are defined in the Caputo sense. In this paper, we investigate the synchronization of the FF hyperchaotic finance system with model uncertainty and external disturbance and propose a new UDE-based control method to realize the synchronization of the FF hyperchaotic finance system. Numerical simulations are carried out to verify the effectiveness and validity of the obtained theoretical results.

Preliminaries.
Firstly, we introduce the definition of the FF-order differential equation in the Caputo sense and some preliminaries of fractional-order chaotic systems.
Consider the following FF-order differential equation in the Caputo sense: Definition 1 (see [13]) with order β, then the FF-derivative of f(x) of order α in the Caputo sense with the power law is given as en, some properties of fractional calculus are introduced.
Property 1 (see [35]). e fractional-order calculus defined by Caputo is a linear operator and satisfies Proof. where λ and μ are real constants.
□ Property 3 (see [36]). Let x ∈ R be a continuous differentiable function, and for any continuous time t ≥ t 0 , i.e.,

Problem Formation.
e FF hyperchaotic finance system is given in the following form: where x ∈ R 4 is the state and u d � Δf(x) + d(t) is the uncertainty and disturbance, i.e., or where u is the controller to be designed. Let system (6) be the master system; then, the corresponding slave system is where e ∈ R 4 is the state, u d and B are given in equation (6), and e main goal of this paper is to design a controller u to meet the following performance:

Main Results
e stabilization of error system (11) with u d � 0 is firstly stabilized by the controller u s , and a conclusion is obtained as follows. Theorem 1. Conside error system (11) can be stabilized, then the controller u s is designed as Proof. Define the following nonnegative function: From Property 1, we get From Property 3, it results in t e 1 � e 1 e 3 − 0.9e 1 + e 4 − e 1 e 2 + x 2 e 1 + x 1 e 2 + u 1 , Mathematical Problems in Engineering 3 Calculating the Caputo derivative of V along the system in equation (15): erefore, master system (6) with u d � 0 synchronizes slave system (9) by the controller u s . en, error system (11) is stabilized, and a result is presented as follows.
□ Theorem 2. Consider error system (11). If (f(x) − f(y), B) can be stabilized and there exists a suitable filter g f (t) such that where (20) and u d satisfies the following structural constraints: where I n is the identity matrix of order n; then, the UDE-based controller u is designed as where u s is given in equation (15), and , ℓ represents Laplace transform, ℓ −1 represents Laplace inverse transform, and * represents convolution.
Proof. Substituting the controller u given in equation (22) into error system (11), we obtain where and the system D α t e(t) � F(x, e) is asymptotically stable according to eorem 1.
According to condition given in equation (19), if the controller u ude meets the following equation then this controller is proposed.
Taking the Laplace transform of both sides of equation (26), it yields that i.e., that is, 0 < α ≤ 1, which completes the proof.

Numerical Simulations
In this section, we use MATLAB to do the numerical simulation of the FF hyperchaotic finance system in the sense of Caputo. Firstly, the numerical simulation of the nominal FF hyperchaotic finance system is carried out. en, the numerical simulation of the FF hyperchaotic finance system with model uncertainty and external disturbance is carried out.

Numerical Simulation of the FF Hyperchaotic Finance System with Model Uncertainty and External Disturbance.
Numerical simulation of the FF hyperchaotic finance system with model uncertainty and external disturbance is carried out. Noted that external disturbance is d(t) two cases are presented as follows.

Conclusions
In conclusion, the synchronization of the FF hyperchaotic finance system with model uncertainty and external disturbances has been investigated. Firstly, a controller has been proposed for the nominal FF hyperchaotic finance system. en, the UDE-based controller has been designed for the FF hyperchaotic finance system. e correctness and validity of the obtained results have been verified by numerical simulation. It is noted that the simulation results show that the aforementioned control method has good performance.
In the future, the obtained control method and the synchronization result are maybe extended to some potential applications, such as the nonlinear digital communication.

Data Availability
No data were used in this paper. Mathematical Problems in Engineering 9