If R 0 < 1, model (2) only has a disease-free equilibrium E 1 ( A d 1, 0 ) ; if R 0 > 1, model (2) has an endemic equilibrium E 2 ( S ∗, I ∗ ) besides E 1 ( A d 1, 0 ), and when for the parameter h are chosen different values, model (2) appears to have many complex and interesting dynamical behaviors.
There are many interesting collective phenomena in complex dynamical networks that can be described by coupled differential equation equations, such as self-organization, synchronization, spatiotemporal chaos and so on.
More complex than the real system, the study on complex dynamical systems has many potential applications in science and engineering.
Because the synchronization of complex dynamical networks can well explain many natural phenomena observed and is one of the important dynamical mechanisms for creating order in complex dynamical networks, the synchronization of coupled dynamical networks has come be a focal point in the study of nonlinear science.
Complex dynamical networks are being studied across many fields of science and engineering today.
Our focus is on dynamical systems with uncountable state spaces, as many complex networks are of this type.
Recently, many authors have studied some new kinds of synchronization for complex dynamical systems, for example, complex complete synchronization [13], complex projective synchronization [14], complex modified projective synchronization [15, 16], and so forth.
In the past decade, many researchers have drawn increasing attention to dynamical analysis of complex dynamical networks due to a variety of their application fields, such as biology, physics, mathematics, sociology and so on [1 6].
During the last two decades, synchronization and control problems of complex dynamical networks have been focused on in many different fields such as mathematics, engineering, social and economic science, etc. [1 8].
