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Synchronization of Complex Dynamical Networks via Event-Triggered Pinning Impulses

Kexue Zhang

Abstract

This article studies the synchronization problem of complex dynamical networks. The impulsive control method is considered with a novel event-triggered pinning algorithm. Sufficient conditions on the network topology are obtained to ensure network synchronization. It is shown that synchronization can be realized with a careful selection of the pinning nodes. Furthermore, an adaptive coupling strength is incorporated into the network to allow network synchronization with an arbitrary selection of the pinning nodes. An example of a network with node dynamics described by the Chen system is studied to demonstrate the theoretical results.

Synchronization of Complex Dynamical Networks via Event-Triggered Pinning Impulses

Abstract

This article studies the synchronization problem of complex dynamical networks. The impulsive control method is considered with a novel event-triggered pinning algorithm. Sufficient conditions on the network topology are obtained to ensure network synchronization. It is shown that synchronization can be realized with a careful selection of the pinning nodes. Furthermore, an adaptive coupling strength is incorporated into the network to allow network synchronization with an arbitrary selection of the pinning nodes. An example of a network with node dynamics described by the Chen system is studied to demonstrate the theoretical results.
Paper Structure (5 sections, 3 theorems, 34 equations, 3 figures, 1 table)

This paper contains 5 sections, 3 theorems, 34 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Synchronization Criteria
  4. An Example
  5. Conclusions

Key Result

Theorem 1

Suppose $\alpha_i>V_i(t_0)$ and $0<d_i<1$ for $i=1,2,...,l$, and denote $\gamma=\|K\|$. Synchronization of the network network-pinning can be achieved if where $\Lambda=\gamma I_{N-l}+c \bar{A}$ and $\bar{A}=(\bar{a}_{ij})_{(N-l)\times(N-l)}$ with $\bar{a}_{ij}=a_{l+i,l+j}$. Moreover, network network-pinning does not exhibit Zeno behavior, that is, $\lim_{k\rightarrow \infty} t^{(i)}_k=\infty$ fo

Figures (3)

  • Figure 1: Network topology.
  • Figure 2: Simulations of network \ref{['network-pinning']} with \ref{['examplef']}.
  • Figure 3: Coupling strength $c(t)$ with adaptive law \ref{['adaptivelaw']}.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Remark 1
  • Corollary 1
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3