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Yaru Zhang, Yingxue Du, "Synchronization Problem of a Novel Fractal-Fractional Orders’ Hyperchaotic Finance System", Mathematical Problems in Engineering, vol. 2021, Article ID 4152160, 10 pages, 2021. https://doi.org/10.1155/2021/4152160
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.
The 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 [1–6]. 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 [7–18]. The 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 , passive control , and sliding model control . Although scholars have made great efforts in the control of fractional-order 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 [24–26]. It is well known that chaotic systems are very sensitive to parametric and external perturbations. Therefore, 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 [27–34]. Therefore, 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.
2. Preliminaries and Problem Formation
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 ). Let be differentiable in opened interval ; if is fractal differentiable on with order , then the FF-derivative of of order in the Caputo sense with the power law is given asThen, some properties of fractional calculus are introduced.
Property 1. (see ). The fractional-order calculus defined by Caputo is a linear operator and satisfies
Proof. where are real constants.
Proof. where is an -norm and is a positive real number.
Property 3. (see ). Let be a continuous differentiable function, and for any continuous time , i.e.,
2.2. Problem Formation
The FF hyperchaotic finance system is given in the following form:where is the state and is the uncertainty and disturbance, i.e.,orwhere is the controller to be designed.
Let system (6) be the master system; then, the corresponding slave system iswhere
The error system is presented aswhere is the state, are given in equation (6), and
The main goal of this paper is to design a controller to meet the following performance:
3. Main Results
The stabilization of error system (11) with is firstly stabilized by the controller , and a conclusion is obtained as follows.
Theorem 1. Conside error system (11) with . If can be stabilized, then the controller is designed as
Proof. Define the following nonnegative function:From Property 1, we getFrom Property 3, it results inCalculating the Caputo derivative of along the system in equation (15):Therefore, master system (6) with synchronizes slave system (9) by the controller .
Then, error system (11) is stabilized, and a result is presented as follows.
Theorem 2. Consider error system (11). If can be stabilized and there exists a suitable filter such thatwhereand satisfies the following structural constraints:where is the identity matrix of order ; then, the UDE-based controller is designed aswhere is given in equation (15), and, , , , represents Laplace transform, represents Laplace inverse transform, and represents convolution.
Proof. Substituting the controller given in equation (22) into error system (11), we obtainwhere . Note thatand the system is asymptotically stable according to Theorem 1.
According to condition given in equation (19), if the controller meets the following equationthen this controller is proposed.
Taking the Laplace transform of both sides of equation (26), it yields thati.e.,Furthermore, we obtainthat is,, which completes the proof.
4. 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. Then, the numerical simulation of the FF hyperchaotic finance system with model uncertainty and external disturbance is carried out.
4.1. Numerical Simulation of the Nominal FF Hyperchaotic Finance System
Numerical simulation is carried out by choosing the initial conditions of master system (6): , the initial conditions of slave system (9): , and . Figure 1 shows that error system (11) is asymptotically stable, that is to say, master system (6) and slave system (9) realize complete synchronization. The states of master system (6) and slave system (9) are displayed in Figure 2, respectively.
4.2. 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 two cases are presented as follows.
When the external disturbance is constant, i.e.,
Numerical simulation is carried out with the initial conditions of master system (6) and slave system (9): and , respectively, and . Figure 3 displays that error system (11) is also asymptotically stable, which implies that master system (6) and slave system (9) reach complete synchronization. The states of master system (6) and slave system (9) are shown in Figure 4, respectively. Figure 5 demonstrates that asymptotically tends to . Figure 6 demonstrates that asymptotically tends to .
The other case is given as follows:
Numerical simulation is carried out with the initial conditions of master system (6) and slave system (9): and , respectively, and . Figure 7 displays that error system (11) is also asymptotically stable, which implies that master system (6) and slave system (9) reach complete synchronization. The states of master system (6) and slave system (9) are shown in Figure 8, respectively. Figure 9 demonstrates that asymptotically tends to . Figure 10 demonstrates that asymptotically tends to .
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. Then, the UDE-based controller has been designed for the FF hyperchaotic finance system. The 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.
No data were used in this paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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Copyright © 2021 Yaru Zhang and Yingxue Du. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.