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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.

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 [

In 1990, Pecora and Carroll [

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 [

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

Thus the master system is in the form

The slave system becomes

Consider the slave system synchronizing with the master system with certain function constraint

The identical synchronization can be as a special case (

Consider the master system to be independent. With a control law, the slave system is discontinuous and becomes

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

for

for

for

for

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

The corresponding boundaries are defined as

The intersection point of the boundaries

Similar to the usual illustration in the discontinuous dynamical systems, the subdomains and boundaries are illustrated in Figures

Subdomains and boundaries of controlled slave system in absolute coordinates.

Separated illustrations for the two boundaries.

Velocity boundary

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 boundary flow is controlled by the master system, and the boundaries change with times. The corresponding dynamical systems on the boundaries are

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

Separated illustrations for the two boundaries in the relative coordinate.

Relative velocity boundary

Relative displacement boundary

The velocity and displacement boundaries in the relative coordinates are constant.

The controlled slave system in relative coordinates becomes

The dynamics on the boundary can be written as

The synchronization of the two systems under the sinusoidal constraint will be discussed. The

The corresponding

A flow sliding on the boundaries of

A flow passing through the boundaries of

A flow grazing the boundaries of

The onset of a sliding flow on the boundaries of

The vanishing of a sliding flow from the boundaries of

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

From (

The synchronization conditions in (

Let

If

Consider a small neighborhood of

From the foregoing equation, the initial point

The conditions of synchronization vanishing for the controlled slave system with

The conditions for onset of synchronization for the controlled slave system with

The authors declare that there is no conflict of interests regarding the publication of this paper.

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).