Abstract
A technique which increased the dimension of slave system is adopted for robust synchronization of chaotic systems with unknown phase difference. The phase difference plays a great role in variation of dynamic behavior of the coupled systems. The phase difference of the sinusoidal forcing term is always assumed to be known in the majority of the existing literature. However, unknown parameter error value has always existed in real problems. This method uses the properties of the triangular function and increases the number of dimensions in the slave system to match the phase of forcing term in the master system. Numerical simulations show that the flexible control variable we first investigated is very important. We expect our results to be of some broader applicabilities.
1. Introduction
Chaos control and synchronization has been attracting more and more interest since the pioneering work of Pecora and Carroll due to its many potential applications in secure communication, information processing, power converters, chemical reactions, artificial neural networks, and so forth, [1].
The research on synchronization has intensively focused on typical autonomous chaotic systems, for example, Chua’s circuit, Chen systems, and Lorenz oscillator [2–5]. Recently, more and more nonautonomous oscillators have become interesting topics, following their wild existence in engineering and science. Many nonautonomous chaotic systems always include the external sinusoidal force terms, such as Duffing oscillators [6], Ueda equations [7], a two-degrees-of-freedom twin-tail system [8], the generalized van der Pol system of nonautonomous form, [9] and so on. In addition, the use of periodically forced chaotic systems in synchronization has advantages of being far less sensitive to noise than autonomous systems [10]. For synchronization of nonautonomous chaotic oscillators with sinusoidal forcing term, the phase and frequency of two systems are often not in coincidence and desynchronize the systems [6].
Recently, the role of phase difference of the externally driven forces has received increasing attention. Yin et al. considered the phase effect of the two mutually coupled Duffing oscillators [6]. Kyprianidia and Stouboulous studied the dynamics of nonlinear system which consists of three identical resistively coupled nonlinear and nonautonomous electric circuits forming a ring in a neural-type connection [11]. Kusumoto and Ohtsubo [12] numerically investigated regions of chaos synchronization in the phase space of the frequency detuning between transmitter and receiver lasers and the optical injection rate in the systems of semiconductor with optical feedback. Nayak and Kuriakose considered the effect of the phase difference of applied fields on the dynamics of mutually coupled Josephson junctions [13].
To our best knowledge, there are two methods of correcting phase difference. The first one only relies on the effect of various kinds of resistive coupling, and investigated the synchronization effects with assumed phase difference and the assumed amplitude of the sinusoidal forcing term. Most literatures used this method. In some instances, unknown parameter value or unknown parameter error value may be more practical. The second one is that Carroll and Pecora used a phase correction circuit to match the phase of the forcing term in master circuit [10].
In the present work, we adopted the incremental dimension technique to focus on the unknown phase difference of the externally driven forces for robust synchronization of chaotic systems. The method shows how to use the slave system of increased dimensions, to imitate the whole forcing term in the master system. The flexible control variable we first investigated plays a great role in imitating results. Numerical simulations are given to explore the relationship among the dynamical states, flexible control variable, and the coupling parameter .
2. Main Result
At first, generalized nonautonomous systems are introduced as follows: where is the state-space vector, is the function space, is the externally driven force, and are unknown parameters.
A traditional slave system is given by where and are any given parameters to estimate the .
The frequency detuning in the response system would be regarded as the special case of phase difference. It is not difficult to find that the sinusoidal forcing term has the following portraits: Inspired by this portraits, we can replace with . The value of would be any given scalar and regarded as flexible control parameter in the slave system. In other words, taking advantage of the portraits of sinusoidal forcing term that the triangular function is proportional to the second derivative, the increased item is used to estimate the whole of the sinusoidal forcing term.
The response system could be designed in the following form: Looking at the increased-dimension slave system, one could wonder whether it is possible to exhibit the similar chaotic behavior of the master system, what happens when such systems interact. The following section explores the relationship among the dynamical states, different flexible control variable , and the coupling parameter by numerical simulation analysis.
3. Horizontal Platform System
3.1. Description of the Horizontal Platform System Model and Equations
The state equations describe the dynamics of horizontal platform system as follows: where are the inertia moment of the platform for three axises, respectively, which penetrate the mass center of the platform, is the damping coefficient, is the proportional constant of the accelerometer, is the acceleration constant of gravity, is the rotation of the platform relative to the earth, is the radius of the Earth, and is harmonic torque.
The horizontal platform system can be represented as the nonautonomous form: where , , , . Equation (6) is considered as a master system.
The traditional slave system is given by and the corresponding slave system would be
For simulation, the values of the system parameters were fixed as , , , , , and . The initial state of the master system is chosen as , and the initial state of the slave system is chosen as .
Remark 1. The systems have been carefully studied by the following authors. Ge et al. [14] had studied the phase effect in unidirectional chaos synchronization of them and investigated the bifurcation diagrams and Lyapunov exponent for phase difference between 0 and . Wu et al. obtained the sufficient synchronization criteria by Lyapunov’s stability theory and estimated the corresponding synchronization error bound [15].
Remark 2. The external forcing item could be regarded as , and have nothing to do with the incremental dimension technique. Therefore, (4) are suitable for both sin and cos type of forcing item.
3.2. Synchronization Analysis of the Horizontal Platform System
In order to estimate the control functions, we define the state errors between the nonautonomous system (6) and autonomous system (8) as , . We get the error system as follows which helps to determine the stability boundaries of the synchronization process. The phase difference is defined as The average value of Euclidean distance is defined as and is the length of the state vector.
The is monitored with different values of flexible control variable and is shown in Figure 1. It indicated that would be the critical value with which the synchronization error will be converged and stabilized in the smallest area. The phase portrait, the error, and similarity of unidirectional coupled systems are shown in Figures 2, 3, and 4. The parameters are set as follows: , in Figure 2, , in Figure 3, , and in Figure 4.
From Figure 1, one can find that the is small when is in the range , which we called critical range for different values of coupling parameter. Moreover, we do similar studies on Duffing system in which the critical range of is . However, we have to find the good value of depending on experiments.
From Figures 2 and 3, one can easily find the similar analysis as Figure 1, the choice of the value of the flexible control variable is very important. It shows that in one case (Figure 3) with the value of in the critical range and the coupling parameter is bigger, the synchronization error bound may be smaller than in other case (Figure 4) with the value of far beyond critical range when the coupling parameter is smaller enough.
Both Figures 5 and 6 are the largest Lyapunov exponent diagrams for the error system by the method of small datasets of Rosenstein et al. [16]. The parameters, such as delay times and embedding windows, are needed in the small datasets method which is determined by the C-C method [17]. Figure 5 indicates that the better synchronization results are obtained under when the coupling parameter value is fixed. In Figure 6, the largest Lyapunov exponent transverse the zeros value from positive to negative when . It can be seen that the error system approaches to a limit cycle when .
Remark 3. Zero crossing of Lyapunov exponent is widely used as a criterion of chaos synchronization. The small data method is a popular method to calculate the largest Lyapunov exponent for its speed, ease of implementation, and robustness to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. However, the realization of Lyapunov exponent needs numerical calculation for infinite evolution time; therefore, this method is not complete in practice [18]. By the way, it is difficult to estimate the critical value of flexible control or to use the Lyapunov direct method, since the error system is not a pure function of state error, especially when the feedback control is a linear states error.
As we known, with , , synchronization of the master system (6) and the slave system (7) is identical except for their initial states. In Figure 7, curve 2 displayed that with , the complete synchronization of two coupled systems can be easily obtained. All the state variables of synchronization error between system (6) and system (7) (curves 2–4) and between system (6) and system (8) (curve 1) converges on the decreasing coupling parameter . Curve 1 is the nearest one with curve 2 which indicates the effectiveness of our dimension expansion approach.
Remark 4. The coupling term dose carry enough information to correct the value of phase difference. For autonomous continuous systems and nonautonomous continuous systems without phase difference, the synchronization would be achieved when the coupling parameter is below a certain critical value. Wu et al. introduced a new definition of global synchronization with error bound, and all of their simulations have shown that the master-slave systems with phase difference hardly achieve the complete (zero error) synchronization even though the feedback coupling parameters are chosen to be smaller enough [15].
The level of mismatch of chaotic synchronization can be given quantitatively by taking the similarity function as a time-averaged difference between the variables and taken with the time drift [19] In Figure 8, the minimum of indicates the existence of some characteristic time shift between system (6) and system (8) (curve 1), as same as system (6) and system (7) (curves 2–5). For smaller phase difference, the minimum of continuously decreases (curves 2–5). The minimum of (curves 1–3) appears to be zero. For more clarity, we enlarged the part of Figure 8 to obtain Figure 9, and get Figures 10 and 11 to correspond to Figures 8 and 9 with different values of coupling parameter .
To sum up, all the simulations above in different aspects show that the better synchronization results would be obtained when the flexible control variable is in the critical range with different values of the coupling parameter.
4. The Ueda Equations with Noise
4.1. Description of Ueda Equations System Model and Equations
The Ueda oscillator is one of the most classical nonautonomous dynamical models and its various features in response, bifurcation, chaos and chaos control have focused considerable attentions, see [20–22]: where is the noise term, is a parameter specifying the intensity of the noise, and is a random variable chosen to be uniformly distributed in the interval . For all the simulations, the values of the system parameters were fixed as , , , , and . The initial state of the master system is chosen as , and the initial state of the slave system is chosen as .
The corresponding response system would be
4.2. Synchronization Analysis of Ueda Equations
Figure 12 indicates that better synchronization results are obtained when and the coupling parameter value is fixed at . In Figure 13, the largest Lyapunov exponent transverse the zeros value form positive to negative when is near zero.
For simplicity, other similar results are omitted here. From the above simulation results, we can see that the critical range of flexible control variable are totally different for different systems. It is an open range for more exploration to mathematic theory about the interesting phenomenon in the future.
5. Conclusions
The phase of forcing term in the slave system may not be the same as that in the master system. We proposed a method, which uses the properties of the triangular function and increase the number of dimensions in the slave system to estimate the phase of forcing term in the master system, while the value of phase difference was assumed to be known at first and its effect of chaos synchronization was investigated in majority of the existing literature. It is noticeable that the flexible control variable we first investigated plays a great role in imitation results. Numerical simulations are performed to support the accuracy of the analytical method. However, the main results of the paper are obtained by numeric simulation analysis, and it lacks strong mathematic theory support. It is an open range for more exploration in the future.
Acknowledgments
This paper has been funded by Project no. CDJXS10181131, supported by the Fundamental Research Funds for the Central Universities. The authors wish to thank the anonymous reviewers for their valuable comments and helpful suggestions which greatly improved the paper’s quality.