^{1,a)}and Yu-Hao Liang

^{1,b)}

### Abstract

Global synchronization in complex networks has attracted considerable interest in various fields. There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach (MMA) proposed by Chen, although having a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. Our approach fixes all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes.

Synchronization of networks of dynamical systems is frequently observed in nature and technology.

^{1,2}Recently, the study of synchronization phenomena in complex networks with different topologies has received much attention.

^{3–15}There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach (MMA) proposed by Chen, despite a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. In the current work, generalizing our previous work,

^{26}which considered time-independent networks, we are able to fix all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes.

We thank referees for suggesting numerous improvements to the original draft.

I. INTRODUCTION

II. BASIC FRAMEWORK

III. MATRICES OF THE COORDINATE TRANSFORMATION

IV. MAIN RESULTS

V. APPLICATIONS AND COMPARISONS

VI. CONCLUSIONS

### Key Topics

- Network topology
- 20.0
- Synchronization
- 19.0
- Oscillators
- 15.0
- Networks
- 11.0
- Chaotic systems
- 8.0

## Figures

The matrix measures of and , with being given in Eq. (11) and are, respectively, represented by the solid line and the dotted lines above. Lines for , are coincided since is circular for all .

The matrix measures of and , with being given in Eq. (11) and are, respectively, represented by the solid line and the dotted lines above. Lines for , are coincided since is circular for all .

The matrix measures of and , with given in Eq. (13) and are, respectively, represented by the solid line and the dotted lines above.

The matrix measures of and , with given in Eq. (13) and are, respectively, represented by the solid line and the dotted lines above.

The matrix measures of and , with being given in Eq. (15) and are, respectively, represented by the solid line and the dotted lines above.

The matrix measures of and , with being given in Eq. (15) and are, respectively, represented by the solid line and the dotted lines above.

Coupling topologies: (a) generalized wheel-typed coupled network with and (b) prism-typed coupled network with . Networks (a) and (b) appear in Examples 6 and 7, respectively.

Coupling topologies: (a) generalized wheel-typed coupled network with and (b) prism-typed coupled network with . Networks (a) and (b) appear in Examples 6 and 7, respectively.

The difference of components of the first two coupled oscillators: (a) the -component partial-state coupling addressed in Case 1 and (b) the -component partial-state coupling addressed in Case 2. In both cases and the outer coupling matrix is given as in Eq. (9).

The difference of components of the first two coupled oscillators: (a) the -component partial-state coupling addressed in Case 1 and (b) the -component partial-state coupling addressed in Case 2. In both cases and the outer coupling matrix is given as in Eq. (9).

The difference of components of the first two coupled oscillators considered in Case 4. Here the -component partial-state coupling is considered with and the outer coupling matrix given as in Eq. (15).

The difference of components of the first two coupled oscillators considered in Case 4. Here the -component partial-state coupling is considered with and the outer coupling matrix given as in Eq. (15).

## Tables

The table gives the matrix measures of , with various size of , which is given in Eq. (9). Since is a circular matrix, the matrix measures of with respect to and are equal. Note that the matrix measure of is , , which is negative regardless of the size of .

The table gives the matrix measures of , with various size of , which is given in Eq. (9). Since is a circular matrix, the matrix measures of with respect to and are equal. Note that the matrix measure of is , , which is negative regardless of the size of .

The table gives the matrix measures of , with various size of , which is given in Example 3.

The table gives the matrix measures of , with various size of , which is given in Example 3.

The table gives the matrix measures of , and , with various size of , which is given in Eq. (18).

The table gives the matrix measures of , and , with various size of , which is given in Eq. (18).

The table gives the matrix measures of , and , with various size of , which is given in Example 7.

The table gives the matrix measures of , and , with various size of , which is given in Example 7.

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