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Explosive synchronization (ES), as one kind of abrupt dynamical transitions in nonlinearly coupled systems, has become a hot spot of modern complex networks. At present, many results of ES are based on the networked Kuramoto oscillators and little attention has been paid to the influence of chaotic dynamics on synchronization transitions. Here, the unified chaotic systems (Lorenz, Lü and Chen) and R össler systems are studied to report evidence of an explosive synchronization of chaotic systems with different topological network structures. The results show that ES is clearly observed in coupled Lorenz systems. However, the continuous transitions take place in the coupled Chen and Lü systems, even though a big shock exits during the synchronization process. In addition, the coupled R össler systems will keep synchronous once the entire network is completely synchronized, although the coupling strength is reduced. Finally, we give some explanations from the dynamical features of the unified chaotic systems and the periodic orbit of the R össler systems.

Complex networks are ubiquitous in the world, such as transportation networks, Internet, wireless networks and phone networks. In 1998, Watts and Strogatz presented the small-world network model [

Recently, it has been shown that discontinuous transitions can take place in networks of periodic oscillators [

Motivated by the above discussions, this paper investigates explosive synchronization in complex dynamical networks coupled with chaotic systems. The chaotic systems to be studied contain the unified chaotic systems (Lorenz, Lü and Chen) [

The rest of this paper is organized as follows. Section 2 introduces the network model and preliminaries. Sections 3 presents lots of simulations of the coupled unified chaotic systems and analyzes the results from the dynamical behaviors of systems. Section 4 provides some simulations of coupled Rössler systems and discusses the phenomenon from the view of the periodic orbit. Finally, Section 5 gives the conclusion of this paper.

Consider an undirected and unweighted network of N coupled chaotic oscillators as follows:

x ˙ i = F ( x i ) + λ ∑ j = 1 N c i j x j , (1)

where i = 1 , 2 , 3 , ⋯ , N , x i ∈ R m is the m-dimensional state variable of the node i and F ( · ) : R m → R m is a continuous vector function. The outer-coupling matrix ( c i j ) N × N is defined as C = A − D and describes the coupling topology of the network, where A = ( a i j ) N × N is the adjacency matrix of the network ( a i j = 1 if nodes i and j are connected, and 0 otherwise), D = ( d i j ) is the diagonal matrix with d i i = ∑ j = 1 N a i j and λ is the coupling strength.

Since the inner coupling matrix is the identity matrix, the synchronization region is unbounded based on the master-stability-function approach [

The global order parameter p ( t ) is described in [

p ( t ) = 2 N ( N − 1 ) ∑ j = 1 N − 1 ∑ i = j + 1 N θ ( r − ‖ x i ( t ) − x j ( t ) ‖ 2 ) , (2)

where θ is Heaviside function

θ ( x ) = { 0 , x ≤ 0 1 , x ≥ 1 , (3)

and r is a small positive constant. Obviously, p ( t ) ∈ [ 0 , 1 ] , can quantify the degree of synchronization among the N oscillators and measure the coherence of the collective motion. The network (1) is fully synchronized when the value p = 1 , while the network (1) is the incoherent state when the value p = 0 .

The local error parameter is described as

e i ( t ) = ‖ x i − x ¯ ‖ , (4)

where i = 1 , 2 , ⋯ , N and x ¯ = ∑ i = 1 N x i / N . The e i ( t ) reflects the motion state of each node in some extent and can measure whether the i-th node is synchronized. The network (1) is fully synchronized when the value e i ( t ) = 0 for all nodes i = 1 , 2 , ⋯ , N .

In this paper, the unified chaotic systems (Lorenz, Lü, Chen) and Rössler systems are selected to study explosive synchronization of coupled chaotic systems in different network structures.

The unified chaotic system is described by [

{ x ˙ = ( 25 α + 10 ) ( y − x ) y ˙ = ( 28 − 35 α ) x − x z + ( 29 α − 1 ) y z ˙ = x y − 1 3 ( α + 8 ) z , (5)

where α is a parameter. The system is chaotic when the parameter α ∈ [ 0 , 1 ] . The unified chaotic system is essentially the convex combinations of Lorenz system and Chen system. It represents the whole family of infinitely many chaotic systems in the middle, while the Lorenz system and the Chen system are just two extreme cases. According to the value of parameter, the system (5) can be classified as follows: when 0 ≤ α < 0.8 , the system (5) belongs to the generalized Lorenz system; when α = 0.8 , the system (5) belongs to the generalized Lü system; when 0.8 < α ≤ 1 , the system (5) belongs to the generalized Chen system.

The Rössler system is described by [

{ x ˙ = − ( y + z ) y ˙ = x + a y z ˙ = b + z ( x − c ) , (6)

where a = 0.15 , b = 0.2 , c = 5.7 .

In further exploring the influence of topological structures on explosive synchronization, we use a one-parameter family of complex networks in [

In this section, a lot of numerical simulations of the unified chaotic systems (Lorenz, Lü, and Chen) are presented in different network structures.

For each panel in

Thus, the first-order transition of the networked Lorenz systems corresponds to a process in which no microscopic signal of synchronization is observed until the threshold of coupling strength λ f is reached.

The conclusive information conveyed by

However, the synchronization diagrams of coupled Lü and Chen systems are contrasted with the Lorenz system, between which the differences are very apparent, as shown in

In this section, some numerical simulations of the coupled Rössler systems are presented in different networks. The global synchronization diagrams p ( λ ) and local error diagrams e i ( λ ) are computed, similarly to the coupled unified chaotic systems, as shown in

For each panel in

local error parameters e i ( λ ) of nodes are continuously reduced to zero, as the coupling strength λ is increased, which results in a second-order transition.

Importantly, as shown in

In conclusion, we reported the transitions towards synchronization of coupled chaotic systems in different networks. Our results show that the emergence of explosive synchronization mainly depends on the dynamics of chaotic systems, especially the specific dynamical state at which the chaotic systems. We have given here numerical proof that a first-order and irreversible synchronization transition takes place in a network of coupled Lorenz systems, although the network structures are more homogeneous. It has been seen that the correlation between node dynamics and topological heterogeneity is not a necessary condition for such an explosive transition in networked Lorenz systems. But for the

networked Lü and Chen systems, a second-order transition towards synchronization is observed, even in the heterogeneous network structures. Surprised, the synchronous state of the networked Rössler systems would not be destroyed once the network achieves complete synchronization, which is because the synchronized Rössler systems move into a periodic orbit. Our findings extend the possibility of encountering first-order transitions to a larger variety of network topologies. This will help us to apply the uncovered mechanism to practice in the future.

This work is supported by the National Natural Science Foundation of China under Grants 61304164 and 61473338, the National Social Science Foundation of China under Grant 18BTJ025, and the Research Project of Hubei Provincial Department of Education under Grant Q20131107.

The authors declare no conflicts of interest regarding the publication of this paper.

Chen, J., Tu, H. and Zhao, J.C. (2019) Explosive Synchronization in Complex Dynamical Networks Coupled with Chaotic Systems. World Journal of Mechanics, 9, 245-258. https://doi.org/10.4236/wjm.2019.911016