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Master-slave synchronization of chaotic systems with a modified impulsive controller

Guizhen Feng12 and Jinde Cao13*

Author Affiliations

1 School of Automation, Southeast University, Nanjing, 210096, China

2 Department of Mathematics and Physics, Nanjing Institute of Industry Technology, Nanjing, 210023, China

3 Department of Mathematics, Southeast University, Nanjing, 210096, China

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Advances in Difference Equations 2013, 2013:24  doi:10.1186/1687-1847-2013-24


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/24


Received:4 July 2012
Accepted:13 January 2013
Published:29 January 2013

© 2013 Feng and Cao; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates global exponential synchronization of chaotic systems by designing a novel impulsive controller. The novel impulsive controller is a combination of current and past error states, which is a modification of the normal impulsive one. Some global exponential stability criteria are derived for the error system by utilizing the stability analysis of impulsive differential equations and differential inequalities and, moreover, the exponential convergence rate can be specified. An illustrative example is given to show the effectiveness of the modified impulsive control scheme.

Keywords:
impulsive synchronization; chaotic systems; impulsive controller; global exponential synchronization

1 Introduction

Synchronization of chaotic systems has become an active research area because of its potential applications in different industrial areas [1-3]. Communication security scheme is one of the hottest fields based on chaos synchronization. In this secure communication scheme, the message signals are injected to a chaotic carrier in the transmitter and then are masked or encrypted. The resulting masked signals are transmitted across a public channel to the receiver. To recover the message in the receiver, the synchronization between the chaotic systems at the transmitter and receiver ends is required. Since Pecora and Carroll [4] originally proposed the synchronization of the drive and response systems with different initial states in 1990, many synchronization techniques such as coupling control [5], adaptive control [6], feedback control [7], fuzzy control [8], observer-based control [9], etc. have been developed in the literature.

Most recently, the impulsive control techniques have been reported and developed to be an interesting method [10-14]. In addition, Yang and Cao [15] investigated the exponential synchronization of the complex dynamical networks with a coupled delay and impulsive control. Guan et al.[16] derived the synchronization of complex dynamical networks with time-varying delays via impulsive distributed control. In [17], the authors analyzed the robustness of impulsive synchronization coupled by linear delayed impulses. The main ideas of these impulses are to use samples of the state variables of the drive system at discrete moments and to synchronize the response system discretely. Once the error system of the two coupled systems is asymptotically stable, they are said to be synchronized. Generally speaking, these impulses are samples of the state variables of the drive system at current discrete moments to drive the response system. However, we can also design a novel impulse using not only current instantaneous errors, but also the previous time instants of errors. By using such a technique, we can increase the impulse distances and reduce the control cost. Although the idea is relatively well defined in control theory, it brings difficulties and challenges to determine the stability of the impulsive differential equation due to a combination of current and past error states. In [18], the authors investigated the synchronization of hyper-chaotic systems with such a modified impulsive controller. Based on the above discussion, we design a more general impulsive controller than the one in [18] and give a new approach to investigate the synchronization of the drive and response system.

The main contributions of this paper are three-fold: (1) An effective modified impulsive controller is designed for the global exponential synchronization of coupled chaotic systems. (2) Due to the additional integral term of the errors corresponding to each impulse, equipped with the definitions and results, we establish a uniform comparison system for this case and derive a sufficient condition in this paper. (3) Global exponential synchronization of the chaotic systems with the proposed impulsive controller can be simultaneously realized. In other words, by adding the summation term in the error dynamics, one could achieve the same effect by increasing the impulse distance and reducing the control cost.

The outline of this paper is listed as follows. In Section 2, model description and some preliminaries are introduced. In Section 3, based on the stability analysis of impulsive functional differential equations, the criteria for the synchronization are derived. In Section 4, a numerical example is given to illustrate the effectiveness and feasibility of the synchronization criteria. Finally, concluding remarks are made in Section 5.

Notation We list some mathematical notations used throughout this paper as follows. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M1">View MathML</a> denote the n-dimensional Euclidean space and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M2">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M3">View MathML</a> be the Euclidean norm and I be the identity matrix. Denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M4">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M5">View MathML</a> as the maximal and minimal eigenvalues of P, respectively. For a sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M6">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M7">View MathML</a> satisfying <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M8">View MathML</a> , let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M10">View MathML</a> , <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M11">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M12">View MathML</a>.

2 Model description and some preliminaries

A chaos-based communication system usually consists of two chaotic systems at the transmitter and receiver ends, which are called the master system and the slave system. At the transmitter end, the master system is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M13">View MathML</a>

(2.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M14">View MathML</a> is the state variable, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M15">View MathML</a> is a constant matrix, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M16">View MathML</a> is a continuous function.

Generally speaking, all the chaotic systems such as Lorenz system, Chen system, Lü system, and Chua’s circuit can be written in the above form.

At the receiver end, the slave system is written in the following form with an impulsive control scheme:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M17">View MathML</a>

(2.2)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M18">View MathML</a> is a continuous function and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M19">View MathML</a> is the modified impulsive hybrid controller designed as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M21">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M22">View MathML</a> are impulsive control gain matrices to be designed and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M23">View MathML</a> is the Dirac delta function. The impulsive instant sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M24">View MathML</a> satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M25">View MathML</a> , with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M26">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M27">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M28">View MathML</a> be the synchronization error between the states of the master system (2.1) and the slave system (2.2).

Remark 1 The proposed modified impulsive control scheme in [18] utilizes feedback from the error at the current time instant and the errors at the previous time instants, which is quite different from the impulsive controllers in [12-17]. By this modification, one can increase the impulsive distance and therefore reduce the control cost effectively. In this paper, we design a more generally modified impulsive control scheme than the one in [18].

Hence, the slave system with the modified impulsive controller can then be described by the following impulsive differential equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M29">View MathML</a>

(2.3)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M30">View MathML</a> is the ‘jump’ in the state variable at the time instant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M6">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M32">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M33">View MathML</a>. For simplicity, we assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M34">View MathML</a> is left continuous at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M35">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M36">View MathML</a>.

Subtracting (2.1) from (2.3) yields the following error dynamics:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M37">View MathML</a>

(2.4)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M38">View MathML</a>. It is easy to see the master system (2.1) and the slave system (2.2) achieve global exponential synchronization if and only if the trivial solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M39">View MathML</a> is globally exponentially stable in the error system (2.4).

Assumption 1 There exist a positive definite matrix P and constant matrices <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M40">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M41">View MathML</a>

Remark 2 Assumption 1 gives some requirements for the dynamics of the master system and the slave system. If the functions describing the master and slave systems satisfy <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M42">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M43">View MathML</a>, one can choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M44">View MathML</a> to satisfy Assumption 1. In addition, several groups of chaotic systems such as Lorenz system, Chen system, Lü system, and Chua’s circuit also satisfy Assumption 1 with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M45">View MathML</a>.

Definition 1 ([19] Average impulsive interval)

The average impulsive interval of the impulsive sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M46">View MathML</a> is less than <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M47">View MathML</a> if there exist a positive integer <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M48">View MathML</a> and a positive number <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M47">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M50">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M51">View MathML</a> denotes the number of impulsive times of the impulsive sequence ζ in the time interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M52">View MathML</a>.

Definition 2 The error dynamical system (2.4) is said to be globally exponentially synchronized if there exist <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M53">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M54">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M55">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M56">View MathML</a>

holds for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M57">View MathML</a> and any initial value.

We will need the following lemmas.

Lemma 1 (see [20])

For any vectors<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M58">View MathML</a>and a positive-definite matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M59">View MathML</a>, the following matrix inequality holds: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M60">View MathML</a>.

Lemma 2 (see [21])

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M61">View MathML</a>be a positive definite matrix, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M62">View MathML</a>

3 Synchronization criteria

In this section, based on the stability analysis for an impulsive delayed system, some sufficient conditions are derived to ensure the global exponential synchronization for the master system and the slave system.

Theorem 1Suppose that Assumption 1 holds and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M26">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M64">View MathML</a>be the largest eigenvalue of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M65">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M66">View MathML</a>be the largest eigenvalue of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M67">View MathML</a>. If there exist a positive definite matrixPsuch that the discrete system

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M68">View MathML</a>

is globally exponentially stable with decay rate<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M69">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M70">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M71">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M72">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M73">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M74">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M75">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M76">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M77">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M78">View MathML</a>. Then the error system (2.4) is globally exponentially stable with the convergence rate<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M79">View MathML</a>, and hence the slave system (2.2) can achieve global exponential synchronization with the master system (2.1).

Proof

Consider a Lyapunov function in the form of

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M80">View MathML</a>

when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M81">View MathML</a>. The Dini derivative of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M82">View MathML</a> along the trajectory of the error system (2.4) can be obtained as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M83">View MathML</a>

(3.1)

where the first inequality is obtained by Assumption 1 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M84">View MathML</a>.

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M85">View MathML</a>

(3.2)

On the other hand, it follows from (2.4) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M86">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M10">View MathML</a> , that we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M88">View MathML</a>

(3.3)

By Lemmas 1 and 2, we can obtain that

(3.4)

(3.5)

(3.6)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M77">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M78">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M74">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M75">View MathML</a> are utilized.

From (3.4)-(3.6), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M96">View MathML</a>

(3.7)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M71">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M72">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M73">View MathML</a>.

Similar to the proof of Theorem 4.2 in [22], by (3.7), for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M100">View MathML</a>, let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M101">View MathML</a>

(3.8)

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M102">View MathML</a>. Then the system of difference equations obtained above together with (3.7) and (3.8) can be expressed as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M103">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M104">View MathML</a>

(3.9)

Let the comparison system be

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M105">View MathML</a>

(3.10)

Then, by the comparison principle, we can get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M106">View MathML</a>

Thus, by the condition in the theorem, there exists a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M55">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M108">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M109">View MathML</a>.

From (3.8), for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M35">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M100">View MathML</a>, we can get that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M112">View MathML</a>

(3.11)

Hence, by Lemma 2, (3.2) and (3.11), and for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M113">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M100">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M115">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M116">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M117">View MathML</a> be the number of impulsive times of the impulsive sequence ζ in the interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M118">View MathML</a>. Hence, we can obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M119">View MathML</a>

(3.12)

Since the average impulsive interval of the impulsive sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M46">View MathML</a> is equal to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M47">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M122">View MathML</a>

Hence, by (3.12), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M123">View MathML</a>

Thus, the trivial solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M124">View MathML</a> of the error system (2.4) is globally exponentially stable with the convergence rate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M79">View MathML</a>, and hence the slave system (2.2) can achieve global exponential synchronization with the master system (2.1). □

Remark 3 In this paper, a modified impulsive control system is adopted to provide the basis for developing global exponential synchronization between the master system and the slave system, which can reduce the impulsive times and the control cost effectively. In addition, to stabilize the error system (2.4) more effectively, we can also consider that the error at the current time instant and the previous time instants play different roles in the impulsive control system. For example, we can suppose that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M126">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M127">View MathML</a>, where η, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M128">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M129">View MathML</a> are constants, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M130">View MathML</a>. Obviously, it is a special case of Theorem 1.

Remark 4 Note that in the proof of Theorem 1, the concept of an average impulsive interval is employed to prove the global exponential stability for the error system under Assumption 1. By this approach, the requirement on the lower bound and upper bound of impulsive interval is removed in Theorem 1, which is different from the conventional ones in the literature.

Remark 5 If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M131">View MathML</a>, the modified impulsive control scheme is the normal impulsive one, such as in [15-17]. Hence, by Theorem 1, we only need a positive definite matrix P such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M132">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M133">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M71">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M135">View MathML</a>, are the same as in Theorem 1. Then the slave system (2.2) can achieve global exponential synchronization with the master system (2.1). In fact, it can be seen from (3.7) that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M136">View MathML</a> is the impulsive strength of the impulsive signal if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M131">View MathML</a>. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M138">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M133">View MathML</a>, the impulse is beneficial for the error system since the difference is reduced. Thus, the error system can be stable easily with the impulsive control system.

In the following, by using Theorem 1, we give some simple corollaries of Theorem 1.

Corollary 1Suppose the impulsive interval is a positive constant Δ, and the impulsive gain matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M140">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M141">View MathML</a>. If there exists a positive definite matrixPsuch that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M142">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M143">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M144">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M145">View MathML</a>andαis the same as in Theorem 1. Then the error system (2.4) is globally exponentially stable with the convergence rate<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M146">View MathML</a>, and hence the slave system (2.2) can achieve global exponential synchronization with the master system (2.1).

Proof The proof is similar to Theorem 1. □

Corollary 2If there exists a positive constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M147">View MathML</a>such that every root<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M148">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M149">View MathML</a>) of the characteristic polynomial

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M150">View MathML</a>

satisfies<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M151">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M149">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M71">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M72">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M73">View MathML</a>, are the same as in Theorem 1. Then the error system (2.4) is globally exponentially stable with the convergence rate<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M146">View MathML</a>, and hence the slave system (2.2) can achieve global exponential synchronization with the master system (2.1).

Proof In fact, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M157">View MathML</a> is the characteristic polynomial of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M158">View MathML</a> in Theorem 1. Hence, if every root satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M151">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M149">View MathML</a>, there exists a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M69">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M162">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M149">View MathML</a>, then the spectral radius of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M158">View MathML</a> satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M165">View MathML</a>. Thus, we conclude that this corollary is true. □

4 Numerical example

In this section, the chaotic system used in this example and simulation is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M166">View MathML</a>

(4.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M168">View MathML</a> are parameters and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M169">View MathML</a> represents the piecewise-linear function of the Chua diode, which is given by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M170">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M171">View MathML</a> are two constants. When <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M172">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M173">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M174">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M175">View MathML</a>, the Chua system is chaotic. We can obtain the double scroll attractor shown in Figure 1 with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M176">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M177">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M178">View MathML</a>.

thumbnailFigure 1. A double scroll attractor in Chua’s circuit.

The Chua oscillator can be written in the form of (2.1), i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M179">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M180">View MathML</a>

The function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M181">View MathML</a> is defined as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M182">View MathML</a>

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M183">View MathML</a>

which shows that Assumption 1 holds with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M184">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M185">View MathML</a>.

Suppose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M140">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M141">View MathML</a>. We should choose proportional and integral gains <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M188">View MathML</a> to satisfy the conditions in Corollary 1. Set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M189">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M190">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M191">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M192">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M193">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M194">View MathML</a>, and an impulsive interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M195">View MathML</a>, one obtains <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M196">View MathML</a> which results in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M197">View MathML</a>. Based on Corollary 1, the error system is globally exponentially stable with the convergence rate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M198">View MathML</a>, and hence the slave system can achieve global exponential synchronization with the master system. The quantity <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M199">View MathML</a> is used to measure the quality of synchronization errors of drive-response dynamical systems, which is simulated in Figure 2.

thumbnailFigure 2. The error system with the impulsive interval<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M200">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M201">View MathML</a>.

To illustrate the effectiveness of the synchronization scheme with the modified impulsive controller, using the given parameters in the original impulsive method <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M189">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M203">View MathML</a>, one obtains <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M204">View MathML</a> according to Corollary 2, which is simulated in Figure 3.

thumbnailFigure 3. The error system with the impulsive interval<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M200">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/24/mathml/M206">View MathML</a>.

The effectiveness of the proposed impulsive controller can be observed from the numerical simulations. This implies that by adding the summation term in the error dynamics, one could reduce the synchronization time with the same impulsive distance. In other words, by adding the summation term in the error dynamics, one could achieve the same effect by increasing the impulse distance and reducing the control cost.

5 Conclusions

This paper is focused on the global exponential synchronization of chaotic systems with an effective modified impulsive controller. Because the modified impulsive controller is a combination of current and past error states, we establish a uniform comparison system for this case and derive a sufficient condition in Theorem 1. At the same time, a numerical example is given to illustrate the effectiveness and feasibility of the proposed methods and results. In other words, by adding the summation term in the error dynamics, one could achieve the same effect by increasing the impulse distance and reducing the control cost.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referee and the associate editor for their very helpful suggestions. This work was jointly supported by the National Natural Science Foundation of China under Grants Nos. 61272530 and 11072059, and the Jiangsu Provincial Natural Science, Foundation of China under Grants No. BK2012741.

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