Iterative Learning and Fractional Order Control for Complex SystemsView this Special Issue
Dynamics Feature and Synchronization of a Robust Fractional-Order Chaotic System
Exploring the dynamics feature of robust chaotic system is an attractive yet recent topic of interest. In this paper, we introduce a three-dimensional fractional-order chaotic system. The important finding by analysis is that the position of signal x3 descends at the speed of 1/c as the parameter b increases, and the signal amplitude of x1, x2 can be controlled by the parameter m in terms of the power function with the index −1/2. What is more, the dynamics remains constant with the variation of parameters b and m. Consequently, this system can provide rich encoding keys for chaotic communication. By considering the properties of amplitude and position modulation, the partial projective synchronization and partial phase synchronization are realized with linear control scheme. The distribution map of optimal synchronization region in the control-parameter space is charted by defining the power consumption of controller. Numerical simulations are executed to confirm the theoretical analysis.
Over the past few decades, the dynamics and property of chaotic system have been extensively studied from different points of views as an active topic [1–4]. The investigation of chaos has benefited the exploration of the complex behavior, intrinsic nonlinear structure of natural system, and the construction of chaotic system, as well as practical applications such as secure communications and signal detection [5–9]. Robust chaotic system can usually provide signal-amplitude modulation by controlling one or some of the parameters in the dynamical equations yet keep the Lyapunov exponents and power spectral density invariable [10–14]. Therefore, it is a type of chaotic system with potential applications in synchronization, signal processing, image encryption, chaotic radar, and chaotic communication [10, 12]. However, according to what we know, there is little information and variety about such system reported in the present literatures.
Although fractional calculus has a history of more than 300 years, it was not really applied to physics, life sciences, and psychology until the last decade [15–18]. Fractional calculus can provide an excellent description of hereditary and memory properties in various materials and processes. The advantage of the fractional system is that it holds more degrees of freedom and that contains a “memory” in it. Therefore, it is of universal significance to study fractional dynamics and its applications [19–22]. Synchronization of integer-order chaotic system is investigated extensively and deeply, and many different types and methods have been presented since the landmark work by Pecora and Carroll [23–29]. However, due to the computational complexity of fractional systems, not all synchronization methods of the integer-order system are suitable for the fractional-order one . In the existing synchronization types, projective synchronization can be interpreted as that the state trajectories of the drive and response systems synchronize to a constant proportion factor. This characteristic is usually used to extend binary digital communication to faster M-nary communication [30, 31]. Another synchronization type to be worth mentioning is phase synchronization, in which the controlled chaotic system adjusts the signal frequencies of dynamics to the rhythm of another chaotic system while the amplitudes go on varying in an irregular and uncorrelated fashion . Phase synchronization has been observed in electrochemical oscillators , plasma discharge tubes , coupled HR neurons , fractional differential chaotic systems , etc.
In this paper, we attempt to explore some new dynamics properties of robust chaotic system by constructing a three-dimensional fractional chaotic system and to further consider the synchronization problem. Analysis of the derived system shows that the parameter m can control the signal amplitude of x1, x2 by the power function with the index −1/2 and that the parameter b can control the position of signal x3 at the descent velocity of 1/c. Then, by considering the vital property of amplitude and position modulation, a linear coupling scheme is designed to realize the partial projective synchronization and partial phase synchronization, respectively. Thus, the proportional factors in projective synchronization and the phase feature in phase synchronization are only determined by system parameters, which will improve the security of synchronous communication. The coupling-parameter range for synchronization is derived analytically and can be effectively narrowed down by appropriately selecting the auxiliary parameters. The distribution map of optimal synchronization region in the coupling-parameter space is further evaluated by defining the power consumption of controller. Numerical simulations are shown to further confirm the theoretical analysis.
2. Model of Fractional-Order Chaotic System
2.1. Model Description
The introduced fractional system is written in the form
The corresponding characteristic equation for any equilibrium point can be deduced as
Specially, when selecting the parameter set , the corresponding equilibrium points and characteristic roots are
With the aid of stability theory of commensurate fractional system , the necessary condition for the chaos emergence is . For the parameter set P and the fractional order q=0.96, the 3D phase diagram and Poincaré map with dense dots are shown in Figure 1, revealing that system (1) is indeed chaotic.
2.2. Dynamics Feature of Amplitude and Position Modulation
Our analysis found that the introduced system has the robust chaos of constant Lyapunov exponents with the variation of parameters m and b. More specifically, the parameter m can control some of the signal amplitude of state variables, and the parameter b can control the position of one of the variables with a certain speed. The revealed feature is rare in the current literature and we believe it will stimulate an exploration boom.
Therefore, the control parameter m can modulate the signal amplitude of variables x1, x2 according to , but the signal x3 keeps its amplitude constant.
We substitute the nonzero equilibrium point or into characteristic equation (3) to obtain
Since (6) is irrespective to parameter m, the Lyapunov exponent spectrum remains invariable when m increases. The dynamics feature of amplitude modulation for system (1) is illustrated by bifurcation diagram and Lyapunov exponent spectrum, as shown in Figure 2. In the figure, the bifurcation diagrams are plotted with the local maxima of variables versus control parameter, and the spectrum of Lyapunov exponents is calculated by the wolf method.
When introducing the linear transformation of , , and , we then deduce system (1) to
Therefore, the position of signal x3 descends at the speed of 1/c as the parameter b increases, while the signals x1, x2 keep their amplitude invariable. Similarly, it is known that (6) is also irrespective to parameter b. Thus, the spectrum of Lyapunov exponent remains constant when b varies in field of real number. The phenomenon of position modulation for system (1) is described by bifurcation diagram and Lyapunov exponent spectrum, as shown in Figure 3. And the speed of position modulation influenced by parameter c is interpreted in Figure 4 by the bifurcation diagram and the maximal Lyapunov exponent spectrum, which signifies that a smaller c will give rise to a larger descent velocity.
3. Preliminaries for Synchronization of Fractional Chaotic System
In this section, some concepts and techniques are recalled for the stability analysis of fractional system.
Definition 1 (see ). When function is continuous, strictly increasing, and , it is said to be of class . If and satisfies with , it is said that is of class .
Theorem 4 (Lyapunov stability and uniform stability of fractional system ). Considering the following fractional-order system with the Caputo definition Let z=0 be an equilibrium point of system (10). If there exists a continuous Lyapunov function and a scalar class-K function satisfying and for any , then system (10) is Lyapunov stable at .
Furthermore, if there exists another scalar class-K function such that then the origin of system (10) is said to be Lyapunov uniformly stable.
4. Partial Projective Synchronization of Fractional Chaotic System
4.1. Synchronization Scheme
The partial projective synchronization of the proposed system is studied here by taking the advantage of the property of amplitude modulation.
Let system (1) be as the drive system, and the response system is expressed as where u2 and u3 are the controllers to be determined.
The synchronization errors are set as , , and , by taking the property of amplitude modulation considered. Then we obtain the error dynamical system
Theorem 6. For the drive system (1) and response system (14), the controllers are designed as and . If the coupling parameters k2 and k3 satisfy where p1, p2, and p3 are positive auxiliary parameters to narrow down the values range of the coupling parameters, then the partial projective synchronization is realized with Lyapunov uniformly stability.
Proof. Let us propose the Lyapunov function , which satisfies for , , and . Taking q-order fractional derivative with respect to time t along the trajectories of (15), it yields with It requires for catering to . Thus we haveTherefore, we finally obtain inequalities (16) and (17). This completes the proof.
4.2. Simulation Analysis
The appropriate selection of parameters p1, p2, and p3 can effectively narrow down the values range of the coupling parameters, which is useful in actual synchronization process. However, it is a complicated relation between the coupling strengths (k2, k3) and (p1, p2, p3). To optimize the synchronization scheme, we will evaluate the distribution map of optimal synchronization region in the coupling-parameter space, by coopting the idea introduced by Ma . We first define the power consumption of the synchronous systems as and , respectively. Thus, the power consumption for evaluating the synchronous controller can be defined as
And for the sake of simplicity, we only consider the maximal power consumption of synchronous controller, as below
Therefore, the optimal synchronization region will be the coupling-parameter region (k2, k3) with the minimum .
In the numerical analysis, we choose the parameter values as a=2, b=1, c=1, n=30, m=1, m1=16, and q=0.96 and set the initial values of drive system and the response system as x(0)=(-0.2, 0.1, 0.5), y(0)=(0.6, -0.5, -0.2). The distribution about the maximal power consumption of synchronous controller in the coupling-parameter space (k2, k3) is illustrated in Figure 5. It is known that the power consumption is mainly determined by the coupling parameter k2, and the optimal synchronization region appears alternately as k2 increases. But to get shorter synchronous transition time, larger values of k2 and k3 are more appropriate.
The synchronization result with k2=3.3, k3=0.6 is depicted in Figure 6, which provides the proportional factors , , and . Figure 7 shows the synchronization result with the same proportional factors, when k2=6, k3=8. The comparative analysis shows that larger parameters k2 and k3 will lead to shorter synchronous transition time.
5. Partial Phase Synchronization of Fractional Chaotic System
5.1. Synchronization Scheme
Select the drive system (1) and the following response system:
And when considering the property of position modulation, the errors of partial phase synchronization are expressed as , , and , then the error dynamical system is obtained as
Theorem 7. For the drive system (1) and response system (23), the controllers are designed as , . If the coupling parameters k2 and k3 satisfy the condition where p1, p2, and p3 are positive constants, then the partial phase synchronization is realized with Lyapunov uniformly stability.
Here we skip the proof process for brevity since it resembles the one of Theorem 6.
5.2. Simulation Analysis
Likewise, to effectively evaluate the optimal synchronization region, we consider the distribution of coupling parameters by defining the maximal power consumption of synchronous controller, as follows:
In the numerical analysis, we choose the parameter values as a=2, b=1, c=1, n=30, m=1, b1=12, and q=0.96 and set the initial values of drive system and the response system as x(0)=(-0.2, 0.1, 0.5), y(0)=(0.6, -0.5, -0.2). The distribution map about the maximal power consumption of synchronous controller in the coupling-parameter space (k2, k3) is illustrated in Figure 8. It is found that the distribution of is complicated when k2 is less than 4, and there exists an optimized coupling-parameter region near k2=1. When k2 continues to increase with the condition of , the value of gradually decreases, and finally it keeps in the optimal synchronization region with .
As the examples to explain our analysis, we select two sets of coupling parameters (k2=1, k3=4) and (k2=10, k3=4) to realize the optimal synchronization. It can be found that we can realize the partial phase synchronization of fractional chaotic system with the selected coupling parameters, but larger parameters k2 and k3 will lead to shorter synchronous transition time, as illustrated in Figures 9 and 10. Further, to analyze the influence of c on the difference between various states of synchronized systems, the representative process of phase synchronization when k2=10, k3=4, and c=0.5 is considered, as shown in Figure 11. The graphs in Figures 10 and 11 signify that a smaller c will lead to a larger difference between the system variables of the phase synchronization.
Robust chaotic system is a recent research interest yet a promising candidate for signal processing, image encryption, chaotic radar, and chaotic communication. Therefore, it is worthwhile to explore the dynamics feature of robust chaotic system. In this paper, we introduced a robust fractional-order chaotic system and revealed the significant dynamics of position modulation and amplitude modulation.
The linear control scheme is designed to realize the partial projective synchronization and partial phase synchronization, by considering the property of amplitude and position modulation. The distribution of optimal synchronization region in the control-parameter space is evaluated by searching the minimum power consumption of the linear controller. Numerical experiments are executed to confirm the theoretical analysis. Since the relation of synchronization variables only depends on the system parameters, it is not easy to attack and accurately reconstruct the drive system. Therefore, it could be significant in secure communication for masking signals.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under Grant no. 51577046, the State Key Program of National Natural Science Foundation of China under Grant no. 51637004, the National Key Research and Development Plan “Important Scientific Instruments and Equipment Development” under Grant no. 2016YFF0102200, Equipment Research Project in Advance under Grant no. 41402040301, the Research Foundation of Education Bureau of Hunan Province of China under Grant no. 16B113, and Hunan Provincial Natural Science Foundation of China under Grant no. 2016JJ4036.
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