Abstract
This paper discusses the synchronization of the Van der Pol equation with a pendulum under the sinusoidal constraint through the theory of discontinuous dynamical systems. The analytical conditions for the sinusoidal synchronization of the Van der Pol equation with a periodically forced pendulum are developed. With the conditions, the sinusoidal synchronizations of the two systems are discussed. Switching points for appearance and vanishing of the partial synchronization are developed.
1. Introduction
With the development of science and technology, coordinate systems are extensively used to quantitatively describe the characteristics and behaviors of the nature. Through the coordinate systems, one can understand and improve the nature better. In order to research the complexity of the changing process with time, one often uses a known system to compare the unknown process with time. When one obtains the similarity and differences of the two processes for a time interval, the complexity of the unknown dynamical system can be determined through the known one on the similar part of the time interval. The synchronization is a kind of similarity in a time interval, which means that the synchronization is a basis to understand an unknown dynamical system from the well-known one. For the reason above, the synchronization of the dynamical systems is an important concept for dynamical systems.
The investigation on the synchronization goes back to the 17th century. In 1673, Huygens [1] described the synchronization of two pendulum clocks with a weak interaction. After Huygens, many results and progress were achieved [2]. In recent decades, a number of new types of synchronization have appeared, and the four basic synchronizations of dynamical systems are identical synchronization, generalized synchronization, phase synchronization, and anticipated and lag synchronization and amplitude envelope synchronization. For any synchronization, there is at least one constraint, and such synchronization may experience the asymptotic stability characteristics. This issue can be referred to in Boccaletti [3] and Pikovsky et al. [4].
In 1990, Pecora and Carroll [5] studied the identical synchronization of two systems connected with common signals by using the criterion of the sub-Lyapunov exponents. In the problem, the signals are treated as constraints for the two systems. Carroll and Pecora [6] used the synchronized circuits to simulate the synchronization of chaos. Since then, such efforts induced a lot of attention to developing the control methods and schemes of the synchronization with constraints. In 1992, two methods for chaos control to achieve the synchronization of two chaotic systems were presented by Pyragas [7] with a small time continuous perturbation. On the basis above, Kapitaniak [8] presented the synchronization of two chaotic systems with such methods in 1994. In the same year, Ding and Ott [9] pointed out that the slave system is not necessary to be a replica of part of master systems. Under the directionally coupled constraint, the generalized synchronization of chaos was discussed by Rulkov et al. [10] in 1995. Kocarev and Parlitz [11] presented the idea that the given systems were treated as the active and passive systems. In 1996, Pyragas [12] discussed the weak and strong synchronization of chaos. In 1997, Ding et al. [13] gave a review on the chaotic control and synchronization, and an adaptive synchronization of chaos was presented by Boccaletti. In 2004, Campos et al. [14] described the multimodal synchronization with chaos, and the definition of master-slave synchronization was presented. In 2006, Teufel et al. [15] discussed the synchronization of two flow-excited pendula, and a review on stability of synchronic dynamics was presented by Chen et al. [16]. In 2007 and 2009, Chen discussed the complete and generalized synchronizations of the systems under noise perturbations [17, 18].
From the above discussion of the synchronization, the synchronization of dynamical systems is that the corresponding flows of the dynamical systems are constrained under special constraint for a time interval. When the constraints are treated as constraint boundaries, the theory of discontinuous dynamical systems can be used to the synchronization of dynamical systems. And the form of synchronization is different when the constraints are different. In 2005, Luo [19] developed a theory for discontinuous dynamical systems and got a lot of results [20–24]. In this paper, we will discuss how the Van der Pol equation will be synchronized with a periodically forced pendulum under the sinusoidal constraint. Consider the pendulum to be the master system and the Van der Pol equation to be the slave system. Under the sinusoidal constraint, how the slave system will be synchronized with the master system is investigated. The analytical conditions of the synchronization will be developed.
2. Master and Slave Systems
Consider a periodically exited pendulum as a master system:
Consider the Van der Pol equation as a slave system:
For convenience, the state variables are defined as and the vector fields are defined as
Thus the master system is in the form where
The slave system becomes where
Consider the slave system synchronizing with the master system with certain function constraint
The identical synchronization can be as a special case (). To get the synchronization, the constraint should be inserted:
Consider the master system to be independent. With a control law, the slave system is discontinuous and becomes where
The master system is independent of the slave system, and the flow will not be changed. But the slave system will be controlled by the master system to be synchronized. Under the control, the slave system possesses four regions and will be discontinuous. The controlled slave system becomes(i)for and , (ii)for and , (iii)for and , (iv)for and ,
3. Discontinuous Description
Under the control laws, the Van der Pol equation has four regions with different vector fields, four boundaries with four different vector fields, and an intersection point with one vector field. The intersection point is the synchronization of the Van der Pol equation with the pendulum. Four domains of the Van der Pol equation in phase space are defined as
The corresponding boundaries are defined as
The intersection point of the boundaries in phase space is
Similar to the usual illustration in the discontinuous dynamical systems, the subdomains and boundaries are illustrated in Figures 1 and 2.
(a) Velocity boundary
(b) Displacement boundary
The corresponding domains and boundaries are labeled, and the dashed curves give the two boundaries. The two boundaries of the controlled Van der Pol equation are determined by the displacement and velocity of the pendulum. The intersection point of the two boundaries is labeled by a filled circular symbol.
Based on the previously defined , the corresponding dynamical system of the controlled slave system is defined as where
The boundary flow is controlled by the master system, and the boundaries change with times. The corresponding dynamical systems on the boundaries are where and , with
From the above equation, it can be seen that the flow is controlled by the master system on the boundaries, and that the boundaries change with time. From the systems in the absolute coordinate, it is difficult to develop the analytical conditions. Thus, the relative coordinates are defined as
The domain and boundaries in the relative coordinate become
The subdomains and boundaries in the relative coordinates are illustrated in Figure 3.
(a) Relative velocity boundary
(b) Relative displacement boundary
The velocity and displacement boundaries in the relative coordinates are constant.
The controlled slave system in relative coordinates becomes where with
The dynamics on the boundary can be written as where with
4. Analytical Conditions for Synchronization
The synchronization of the two systems under the sinusoidal constraint will be discussed. The -functions are introduced in the relative coordinates for at where and are the zero-order and first-order -functions of the flow in the domain at the boundary . In this paper, the normal vectors of the boundaries are
The corresponding -functions for the boundary are where
4.1. Flow Switchability on the Separation Boundary
(i)A flow sliding on the boundaries of , and for the controlled system satisfies (ii)A flow passing through the boundaries of , and for the controlled system satisfies (iii)A flow grazing the boundaries of , and for the controlled system satisfies (iv)The onset of a sliding flow on the boundaries of , and for the controlled system satisfies (v)The vanishing of a sliding flow from the boundaries of , and to a domain for the controlled system satisfies
4.2. Synchronization Conditions
With the theory of the switchability of a flow, the conditions for the synchronization of the two dynamical systems at the intersection of the two separation boundaries are
From (27), we define four basic functions:
The synchronization conditions in (41) become
Let . The conditions are where the -function becomes
If , , the first two equations can be satisfied, and the last two equations give the synchronization invariant set; that is,
Consider a small neighborhood of ; the attractive conditions for are given by
From the foregoing equation, the initial point and can be obtained for the system in relative coordinate. Thus the initial conditions for the controlled slave system should be determined by
The conditions of synchronization vanishing for the controlled slave system with are from ; and from ; and from ; and from .
The conditions for onset of synchronization for the controlled slave system with are from ; and from ; and from ; and from .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11171192) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20123704110001).