Volume 16 Number 2
April 2019
Article Contents
Xiu-Li Chai and Zhi-Hua Gan. Function Projective Lag Synchronization of Chaotic Systems with Certain Parameters via Adaptive-impulsive Control. International Journal of Automation and Computing, vol. 16, no. 2, pp. 238-247, 2019. doi: 10.1007/s11633-016-1020-4
Cite as: Xiu-Li Chai and Zhi-Hua Gan. Function Projective Lag Synchronization of Chaotic Systems with Certain Parameters via Adaptive-impulsive Control. International Journal of Automation and Computing, vol. 16, no. 2, pp. 238-247, 2019. doi: 10.1007/s11633-016-1020-4

Function Projective Lag Synchronization of Chaotic Systems with Certain Parameters via Adaptive-impulsive Control

Author Biography:
  • Xiu-Li Chai received the Ph. D. degree in mechatronics engineering from South China University of Technology, China in 2008. She is currently an associate professor in the School of Computer and Information Engineering of Henan University, and is doing postdoctoral research in Geography of Henan University, China.
    Her research interests include nonlinear control and multimedia security.
    E-mail: chaixiuli@henu.edu.cn
    ORCID iD: 0000-0001-5727-8933

  • Corresponding author: Zhi-Hua Gan received the B. Sc. degree in applied mathematics from Henan University, China in 2008. He is now a Ph. D. candidate in the Beijing Institute of Technology, China.
    His research interests include nonlinear control and multimedia security.
    E-mail: gzh@henu.edu.cn (Corresponding author)
    ORCID iD: 0000-0002-2372-2853
  • Received: 2015-04-13
  • Accepted: 2015-12-25
  • Published Online: 2017-06-06
通讯作者: 陈斌, bchen63@163.com
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Function Projective Lag Synchronization of Chaotic Systems with Certain Parameters via Adaptive-impulsive Control

  • Corresponding author: Zhi-Hua Gan received the B. Sc. degree in applied mathematics from Henan University, China in 2008. He is now a Ph. D. candidate in the Beijing Institute of Technology, China.
    His research interests include nonlinear control and multimedia security.
    E-mail: gzh@henu.edu.cn (Corresponding author)
    ORCID iD: 0000-0002-2372-2853

Abstract: A new method is presented to study the function projective lag synchronization (FPLS) of chaotic systems via adaptive-impulsive control. To achieve synchronization, suitable nonlinear adaptive-impulsive controllers are designed. Based on the Lyapunov stability theory and the impulsive control technology, some effective sufficient conditions are derived to ensure the drive system and the response system can be rapidly lag synchronized up to the given scaling function matrix. Numerical simulations are presented to verify the effectiveness and the feasibility of the analytical results.

Xiu-Li Chai and Zhi-Hua Gan. Function Projective Lag Synchronization of Chaotic Systems with Certain Parameters via Adaptive-impulsive Control. International Journal of Automation and Computing, vol. 16, no. 2, pp. 238-247, 2019. doi: 10.1007/s11633-016-1020-4
Citation: Xiu-Li Chai and Zhi-Hua Gan. Function Projective Lag Synchronization of Chaotic Systems with Certain Parameters via Adaptive-impulsive Control. International Journal of Automation and Computing, vol. 16, no. 2, pp. 238-247, 2019. doi: 10.1007/s11633-016-1020-4
  • In 1963, Lorenz discovered the chaotic phenomenon in meteorology, and afterwards many chaotic systems have been presented and studied, such as Chua system, Chen system, Lorenz system, Rossler system and others. As a typical nonlinear system, chaotic systems have a dense collection of points with periodic orbits, are highly sensitive to initial conditions and system parameters, and topologically transitive. And chaotic systems are applied in various fields including mechanics, communications, laser technology, radiophysics and electronics in terms of Chua circuit, chemistry as the Belouzov-Zhabotinski reaction and biochemistry, biology and medicine, economics, fluid dynamics by the Rayleigh-Benard convection and others. Due to these properties, synchronization phenomenon may be found in many physical and biological systems such as heart beat, walking, coordinated robot motion and others.

    Since the concept of chaos synchronization introduced by Pecora and Carroll[1], chaos synchronization has been an active research topic due to its importance in theory and its potential applications in neural networks, biology, secure communication and other fields[2-5]. Chaos synchronization is the state in which the drive system and the response system have precisely identical trajectories for time to infinity, and such synchronization can be regarded as complete synchronization[6] or identical synchronization. So far, several different regimes of chaos synchronization have been developed, such as phase synchronization[7], projective synchronization[8], lag synchronization[9], function projective synchronization[10-12], etc.

    Recently, a new synchronization phenomenon called function projective lag synchronization (FPLS) is introduced, where the states of two chaotic systems are asymptotically lag synchronized up to a desired scaling function matrix[13-16]. This synchronization phenomenon has two outstanding advantages. First, it has proportionality between its synchronized dynamical states, and this can be employed to extend binary digital to M-nary digital communication for achieving fast communication. Moreover, time delay is ubiquitous between communication channels in a real-world situation. Thus, it is natural to consider time delay when we deal with synchronization problem. So, function projective lag synchronization has been a hot topic.

    In recent years, some studies have considered the function projective lag synchronization. Based on Lyapunov method and linear matrix inequality (LMI) framework, Lee et al.[13] studied FPLS of chaotic systems with disturbances using adaptive control method, designed the controllers to guarantee asymptotic stability for error dynamics and tested the effectiveness of the method through numerical simulations. In [14], FPLS in chaotic systems was proposed, a suitable controller was given, and numerical simulations of FPLS between Lorenz system and Lu system and two identical hyperchaotic Chen system were made to verify the effectiveness of the methods. Wu and Lu[15] investigated function projective lag synchronization between two different complex networks with nonidentical nodes and different chaotic systems with fully uncertain parameters. Recently, Wang et al.[16] studied the function projective lag synchronization of fractional-order chaotic systems, a nonlinear fractional-order controller was designed for the synchronization of systems with the same and different dimensions based on the stability theorem of linear fractional-order systems, the FPLS was achieved in both reduced and increased dimensions for different dimensional systems, and FPLS between the same three dimensional fractional-order Rossler system and Lu systems, the four-dimensional fractional-order hyperchaotic Lorenz system and three dimensional fractional-order Rossler system, three dimensional fractional-order Lu system and four-dimensional fractional-order hyperchaotic Chen system were realized.

    Some important control methods including continuous control and discontinuous control have been introduced for stabilizing and synchronizing chaotic systems, such as adaptive control[17-19], sliding mode control[20, 21], impulsive control[22-26], intermittent control[27, 28], fuzzy control[29-31] and others. In recent years, there is an increasing interest in adaptive-impulsive control for synchronization of chaotic systems due to its theoretical and practical significance. Using the adaptive-impulsive control method, only the state information at discrete time instants is needed, which is more efficient, secure, and thus, useful in a great number of applications, particularly for secure communication. Wan and Sun[32] investigated the nonlinear adaptive-impulsive synchronization of chaotic systems, applied it to quantum cellular neural network (quantum-CNN), and found adaptive-impulsive controllers more effective than the adaptive control scheme. Li et al.[33] discussed adaptive impulsive synchronization and parameter identification of a class of chaotic and hyperchaotic systems. In [34], Chen et al. derived an adaptive impulsive control with only one restriction criterion to achieve synchronization of nonlinear chaotic systems in the exponential rate of convergence. Wu and Cao[35] studied function projective synchronization of chaotic systems via nonlinear adaptive-impulsive control. Li et al.[36] proposed a novel method to realize the adaptive impulsive synchronization of two general fractional order chaotic systems with uncertain and unknown parameters, Mittag-Leffter function was applied to the stability theory of fractional order differential equation in the proof, the criteria of system synchronization was established, and adaptive impulsive synchronization between fractional-order Lorenz system (the drive system) and fractional order economic system (the response system) was attained. Gao and Hu[37] studied the adaptive impulsive synchronization and parameter estimation of chaotic systems by using discontinuous drive signals, the corresponding theoretical proof was presented to guarantee the effectiveness of the control strategy, and concrete schemes were designed for achieve the synchronization of quantum cellular neural network and well known Chen chaotic system, and numerical simulations were given to demonstrate the effectiveness of the proposed scheme. Zhang et al.[38] constructed a drive network and a suitable impulsively controlled slave network, obtained some synchronization criteria for the uncertain complex networks based on the adaptive impulsive method, the tracking parameters were also gotten simultaneously, and a simple network with five identical nodes was considered, Lorenz system was regarded as the node's dynamical function, numerical simulations were presented to verify the validity of the proposed synchronization criteria. Wu et al.[39] studied the outer synchronization between drive and response networks via adaptive impulsive pinning control, the controlled nodes were changed adaptively according to the synchronization errors at different impulsive instants, the synchronization criterion was derived based on the Lyapunov function method and mathematical analysis, the impulsive intervals for achieving synchronization of the drive network and the response network were estimated through an algorithm for determining the impulsive instants, and several numerical examples were given to illustrate the theoretical results. To the best of our knowledge, there is no more report on function projective lag synchronization of chaotic systems via adaptive-impulsive control.

    Motivated by the aforementioned analysis, in this paper we first investigate FPLS of chaotic systems by adaptive-impulsive control. By using the Lyapunov stability theory and impulsive control technique, suitable controller is designed to obtain function projective lag synchronization of chaotic systems. An estimate of the upper bound of impulsive instants is presented. Numerical simulations of three examples are provided to show the effectiveness of our theoretical results.

    The rest of the paper is organized as follows. In Section 2, adaptive-impulsive controller is designed and a stability criterion is derived. In Section 3, the concrete nonlinear adaptive-impulsive controllers of a 3D Lorenz chaotic system are given, and then numerical simulations are presented to verify the effectiveness of the method. In Section 4, numerical simulation results of a 4D Qi hyperchaotic system are supplied to illustrate the effectiveness of our results. Section 5 shows the numerical simulation results of the unified chaotic system to verify the generality of our methods. Finally, we draw some conclusions from the present study in Section 6.

  • Consider the drive (master) chaotic system and the response (slave) chaotic system:

    $ \begin{align} &{{\dot x}}\left( t \right) = f\left( {{{x}}\left( t \right)} \right) \end{align} $

    (1)

    $ \begin{align} &{{\dot y}}\left( t \right) = g\left( {{{y}}\left( t \right)} \right) + {{U}}\left( t \right) \end{align} $

    (2)

    where $ {{x}}(t) = (x_1, x_2, \cdots, x_n)^{\rm T} $, $ {{y}}(t) = (y_1, y_2, \cdots, y_n)^{\rm T} \in {{\bf R}_{{n}}} $ are the state vectors of systems (1) and (2), respectively; $ f, g: {{\bf R}^{{n}}} \to {{\bf R}^{{n}}} $ are two continuous vector functions and $ {U}(t) $ is a controller to be designed for synchronization between systems (1) and (2).

    Definition 1. It is said that the synchronization between systems (1) and (2) is FPLS if there exists an effective controller $ {U}(t) $ such that the following equation

    $ \begin{align} \mathop {\lim }\limits_{t \to \infty } \left\| {{{e}}\left( t \right)} \right\| = \mathop {\lim }\limits_{t \to \infty } \left\| {{{\alpha }}(t){{x}}\left( {t - {{r}}} \right) - {{y}}\left( t \right)} \right\| = 0. \end{align} $

    (3)

    holds for any initial conditions $ {{x}}(0) $ and $ {{y}}(0) $, where $ {{\alpha}} \left(t \right){\rm{ }} = {\rm{ diag}}\left({{\alpha _{\rm{1}}}\left(t \right), {\alpha _{\rm{2}}}\left(t \right), \cdots, {\alpha _{{n}}}\left(t \right)} \right), {\alpha _{{i}}}\left(t \right) \ne 0{\rm{ }}\left({i = {\rm{ 1}}, {\rm{2}}, \cdots, {{n}}} \right) $ are continuously scaling function matrix. $ {{e}}\left(t \right) $ is system error vector, and $ r $ is time delay.

    Remark 1. When time delay $ {{r} = 0} $, then function projective lag synchronization becomes function projective synchronization. When time delay $ {{r} = 0} $ and $ \alpha (t) = {\rm{constant}} $, then function projective lag synchronization is projective synchronization. When time delay $ {{r} = 0} $ and $ \alpha (t) = {\rm{1}} $, then function projective lag synchronization is complete synchronization.

    We rewrite the drive system (1) as

    $ \begin{align} {{\dot x}} = {{Ax}} + g({{x}}) \end{align} $

    (4)

    where $ {{x}} = (x_1, x_2, \cdots, x_n)^{\rm T} \in {{\bf{R}}_{{n}}} $ is the state vector, $ {{A}} $ is a constant matrix, $ g: {{\bf{R}}^{{n}}} \to {{\bf{R}}^{{n}}} $ is a continuous function. The corresponding response system under the adaptive-impulsive control is given by

    $ \begin{align} \left\{ \begin{array}{l} {{\dot y}} = {{Ay}} + g({{y}}) + {{u}}, \; \; t \ne {t_k}\\ \Delta {{y}} = {{y}}(t_k^ + ) - {{y}}({t_k}) = \\ \quad \; \; \quad {{{B_k}}}({{\alpha }}(t){{x}}(t - {r}) - {{y}}), \; \; t = {t_k}\\ k = 1, 2, \cdots\end{array} \right.\\[-6mm] \end{align} $

    (5)

    where $ {{y}} = (y_1, y_2, \cdots, y_n)^{\rm T} \in {{\bf{R}}_{{n}}} $ is the state vector and is left continuous at the time instants $ {t_k} $, $ 0 < {t_1} < {t_2} < \cdots < {t_k} < {t_{k + 1}} < \cdots, {t_k} \to \infty $ as $ {k} \to \infty. $ $ {{B_k}} = {\rm{diag}}\left\{ {{B_{{\rm{1}}k}}, {B_{{\rm{2}}k}}, {\rm{ }} \cdots, {B_{ik}}, {B_{\left({i + {\rm{1}}} \right)k}}, {\rm{ }} \cdots } \right\} \in {{\bf{R}}^{{{n}} \times {{n}}}} $ is impulsive control gain constant matrix. $ \{ {{u}}; {{B_k}}, {t_k}\} $ is nonlinear adaptive-impulsive controller to be designed.

    Let $ {{e}}\left(t \right) = {{\alpha }}\left(t \right){{x}}\left({t - r} \right) - {{y}}\left(t \right) $ be the function projective lag synchronization error between systems (4) and (5). The error system is described as

    $ \begin{align} \left\{ \begin{array}{l} {{\dot e}}(t) = {{\dot \alpha }}(t){{x}}(t - r) + {{\alpha }}(t){{\dot x}}(t - r) - {{\dot y}}(t) = \\ \quad \; \; \quad {{Ae}} + {{\dot \alpha }}(t){{x}}(t - r) + \\ {{\alpha }}(t)g({{x}}(t - r)) - g({{y}}(t)) - {{u}}, \ t \ne {t_k}\\ \Delta {{e}} = {{e}}(t_k^ + ) - {{e}}(t_k^ - ) = - {{{B_k}}}{{e}}, \ t = {t_k}\\ \quad \; \; \quad k = 1, 2, \cdots \end{array} \right.\\[-6mm] \end{align} $

    (6)

    Our goal is to achieve FPLS between the drive system (4) and the response system (5) by constructing an effective adaptive-impulsive controller such that $ \mathop {\lim }_{t \to \infty } \left\| {{{e}}\left(t \right)} \right\| = 0 $. To this end, we design the adaptive-impulsive controller as follows:

    $ \begin{align} {{u}} = {{\dot \alpha }}(t){{x}}(t - r) + {{\alpha }}(t)g({{x}}(t - r)) - g({{y}}(t)) - {{le}}. \end{align} $

    (7)

    where $ {{l}} = {\rm{diag}}\{ {l_{\rm{1}}}, {l_{\rm{2}}}, {l_{\rm{3}}}, \cdots \} $ is the undetermined constant matrix.

    By substituting (7) into (6), one can obtain

    $ \begin{align} \left\{ \begin{array}{l} {{\dot e}}(t) = {{Ae}} + {{le}}, \ t \ne {t_k}\\ \Delta {{e}} = {{e}}(t_k^ + ) - {{e}}(t_k^ - ) = - {{{B}}_{{k}}}{{e}}, \ t = {t_k}\\k = 1, 2, \cdots \end{array} \right.\\[-6mm] \end{align} $

    (8)

    Theorem 1. Assume $ \rho $ and $ \mu $ be the largest eigenvalues of $ ({{I}} -{{B}}){({{I}} -{{B}})^{\rm T}} $ and $ \Big({\frac{(A + A^{\rm T})} {2}} + {\frac{(l + l^{\rm T})} {2}} \Big), $ respectively. The response system (5) is synchronized with the drive system (4), if one of the following conditions holds: 1) when $ \mu \ge 0 $, if there exists a constant $ \xi > 1 $, such that

    $ \begin{align} \ln (\xi {\rho _k}) + 2\mu ({t_k} -{t_{k - 1}}) \le 0, \; \; k = 1, 2, \cdots \end{align} $

    (9)

    2) when $ \mu < 0 $, and if there exists a constant $ \zeta \; (0 \le \zeta < - 2\mu) $, such that

    $ \begin{align} \ln ({\rho _k}) - \zeta ({t_k} - {t_{k - 1}}) \le 0, \; \; k = 1, 2, \cdots \end{align} $

    (10)

    Proof. Construct the Lyapunov function in the form of

    $ \begin{align} V(t) = \frac{1}{2}{{{e}}^{\rm T}}{{e}}. \end{align} $

    (11)

    For $ t \in ({t_{k - 1}}, {t_k}], k \in N $, its derivative along the trajectory of error system (8) is

    $ \begin{align} \begin{array}{l} \dot V(t) = \dfrac{1}{2}{{{{\dot e}}}^{\rm T}}{{e}} + \dfrac{1}{2}{{{e}}^{\rm T}}{{\dot e}} = \\\qquad \dfrac{1}{2}{({{Ae}} + {{le}})^{\rm T}}{{e}} + \dfrac{1}{2}{{{e}}^{\rm T}}({{Ae}} + {{le}}) = \\\qquad \dfrac{1}{2}{{{e}}^{\rm T}}{({{A}} + {{l}})^{\rm T}}{{e}} + \dfrac{1}{2}{{{e}}^{\rm T}}({{A}} + {{l}}){{e}} = \\\qquad {{{e}}^{\rm T}}\Big(\dfrac{{{{({{A}} + {{l}})}^{\rm T}} + ({{A}} + {{l}})}}{2}\Big){{e}} = \\\qquad {{{e}}^{\rm T}}\Big(\dfrac{{{{A}} + {{{A}}^{\rm T}}}}{2} + \dfrac{{{{l}} + {{{l}}^{\rm T}}}}{2}\Big){{e}}. \end{array}\\[-6mm] \end{align} $

    (12)

    Equation (12) can be changed to

    $ \begin{align} \dot V(t) = {{{e}}^{\rm T}}\Big(\frac{{{{A}} + {{{A}}^{\rm T}}}}{2} + \frac{{{{l}} + {{{l}}^{\rm T}}}}{2}\Big){{e}} \le \mu {{{e}}^{\rm T}}{{e}} = 2\mu V(t). \end{align} $

    (13)

    On the other hand, for $ t = {t_k} $, we have

    $ \begin{align} V(t_k^ + ) = \, &\frac{1}{2}{{{e}}^{\rm T}}(t_k^ + ){{e}}(t_k^ + ) = \\ & \frac{1}{2}{{{e}}^{\rm T}}({t_k}){({{I}} -{{{B}}_k})^{\rm T}}({{I}} -{{{B}}_k}){{e}}({t_k}) \le \\ & \frac{1}{2}{\rho _k}{{{e}}^{\rm T}}({t_k}){{e}}(t{}_k) \le {\rho _k}V({t_k}). \end{align} $

    (14)

    For $ t \in ({t_0}, {t_1}] $, it may be inferred from inequality (13) that $ V(t) \le V(t_0^ +)\exp (2\mu (t -{t_0})) $, which leads to $ V({t_1}) \le V(t_0^ +)\exp (2\mu ({t_1} -{t_0})) $.

    When $ t = t_1^ + $, one has $ V(t_1^ +) \le {\rho _1}V({t_1}) \le {\rho _1}V(t_0^ +)\exp (2\mu ({t_1} -{t_0})) $.

    Similarly, for $ t \in ({t_1}, {t_2}] $, we have $ V(t) \le V(t_1^ +)\exp (2\mu (t -{t_1})) \le {\rho _1}V(t_0^ +)\exp (2\mu (t -{t_0})) $.

    In general, for $ t \in ({t_k}, {t_{k + 1}}], \; \; k = 1, 2, \cdots $, then

    $ \begin{align} V(t) \le {\rho _1}{\rho _2}\cdots{\rho _k}V(t_0^ + )\exp (2\mu (t - {t_0})). \end{align} $

    (15)

    1) When $ \mu \ge 0 $, if there exists a constant such that $ {\rho _k}\exp (2\mu ({t_k} - {t_{k - 1}})) \le \frac{1}{\xi }, k = 1, 2, \cdots $, then

    $ \begin{align} V(t) \le& {\rho _1}{\rho _2}\cdots{\rho _k}V(t_0^ + )\exp (2\mu (t - {t_0}))\le\\ & V(t_0^ + ){\rho _1}\exp (2\mu ({t_1} - {t_0})){\rho _2}\exp (2\mu ({t_2} - {t_1}))\cdots\\ & {\rho _k}\exp (2\mu ({t_k} - {t_{k - 1}}))\exp (2\mu (t - {t_k}))\le\\ & \frac{1}{{{\xi ^k}}}V(t_0^ + )\exp (2\mu (t - {t_k})). \end{align} $

    (16)

    2) For $ \mu < 0 $, and if there exists a constant $ \zeta \; (0 \le \zeta < - 2\mu) $, such that $ \ln ({\rho _k}) - \zeta ({t_k} - {t_{k - 1}}) \le 0 $, i.e., $ {\rho _k} \le \exp (\zeta ({t_k} - {t_{k - 1}})) $, then

    $ \begin{align} V(t) \le &{\rho _1}{\rho _2}\cdots{\rho _k}V(t_0^ + )\exp (2\mu (t - {t_0})) \le \\ & V(t_0^ + )\exp (\zeta (t - {t_0}))\exp (2\mu (t - {t_0}))\le\\ & V(t_0^ + )\exp ((\zeta + 2\mu )(t - {t_0})). \end{align} $

    (17)

    From inequalities (16) and (17), it can be inferred that $ V(t) \to 0 $ as $ k \to \infty $, which implies that all the errors $ {e_i} \to 0 $. Thus, the error dynamical system (8) is asymptotically stable, which implies that systems (4) and (5) achieve function projective lag synchronization.

    Remark 2. In order to guarantee the boundedness of the impulsive intervals $ ({t_k} - {t_{k - 1}}) $, $ \mu \ge 0 $ should be satisfied. For $ \mu $ is the largest eigenvalues of $ \Big(\frac{A + A^{\rm T}} { 2} + {\frac{l + l^{\rm T}} {2}} \Big) $, then through choosing the proper $ l $, the condition of $ \mu \ge 0 $ may be attained.

    Corollary 1. We assume that the impulses are equidistant and separated by $ \delta $, i.e., for any $ k \in N $, $ {t_k} - {t_{k - 1}} = \delta $. In practice, for convenience, the gains $ {{{B_k}}} $ are always chosen as a constant matrix and assume $ {{{B_k}}} = {{B}} $. If there exists a constant $ \xi > 1 $, such that the following condition holds

    $ \begin{align} \ln (\xi \rho ) + 2\mu ({t_k} - {t_{k - 1}}) \le 0, \; \; k = 1, 2, \cdots \end{align} $

    (18)

    Then, the adaptive-impulsively controlled response system (5) asymptotically synchronizes with the drive system (4), where $ \rho $ and $ \mu $ be the largest eigenvalues of $ ({{I}} - {{B}}){({{I}} - {{B}})^{\rm T}} $ and $ \Big(\frac{A + A^{\rm T}} { 2} + {\frac{l + l^{\rm T}} {2}} \Big) $. Moreover, an estimate of the upper bound of $ \delta $ is given as $ {\delta _{\max }} = - \frac{{\ln (\xi \rho)}}{{2\mu }} $.

    Remark 3. In Theorem 1 and Corollary 1, the adaptive-impulsive synchronization scheme for a class of chaotic or hyperchaotic system is presented, and some effective and practical sufficient conditions are derived to guarantee that the drive system and the response system are synchronized in the sense of FPLS.

    Remark 4. Adaptive controllers have the merit of simple design, but continuous control needs more energy. Impulsive controllers can save much energy. The new adaptive-impulsive controller the paper constitutes integrates the advantages of the adaptive controller and impulsive controller, it is designed simply and wastes less energy. Therefore, the proposed method in this paper can be applied to many fields, particularly for secure communication, and more secure encryption system can be attained.

  • In this section, an example is provided to verify and show the effectiveness of the proposed adaptive-impulsive controller.

  • One Lorenz type system can be presented as the following equation:

    $ \begin{align} \left\{ \begin{array}{l} {{\dot x}_1} = a({x_2} - {x_1})\\ {{\dot x}_2} = b{x_1} + c{x_2} - {x_1}{x_3}\\ {{\dot x}_3} = - d{x_3} + {x_1}{x_2} \end{array} \right.\\[-5mm] \end{align} $

    (19)

    where $ {x_1} $, $ {x_2} $ and $ {x_3} $ are state variables, and $ a $, $ b $, $ c $ and $ d $ are system parameters. When $ a = 35 $, $ b = -7 $, $ c = 20 $, $ d = 3 $, it exhibits chaotic behaviors as shown in Fig. 1.

    Figure 1.  Chaotic attractor

    Selecting the system (19) as the drive system, the response system via adaptive-impulsive controller is given by

    $ \begin{align} \quad\left\{ \begin{array}{l} {{\dot y}_1} = a({y_2} - {y_1}) + {u_1}, \ t \ne {t_k}\\ {{\dot y}_2} = b{y_1} + c{y_2} - {y_1}{y_3} + {u_2}, \ t \ne {t_k}\\ {{\dot y}_3} = - d{y_3} + {y_1}{y_2} + {u_3}, \ t \ne {t_k}\\ \Delta {{y}} = y(t_k^ + ) - y({t_k}) = {{{B_k}}}({{\alpha }}(t){{x}}(t - r) - {{y}}), \ t = {t_k} \end{array} \right.\\ \end{align} $

    (20)

    where $ {{y}}(t) = (y_1, y_2, y_3)^{\rm T} \in {{\bf{R}}_{{n}}} $ is state variables, $ {\alpha } \left(t \right) = {\rm{diag}}\left({{\alpha _{\rm{1}}}\left(t \right), {\alpha _{\rm{2}}}\left(t \right), {\alpha _{\rm{3}}}\left(t \right)} \right) $ is scaling function matrix, r is time delay, $ {{B_k}} = {\rm{diag}}\left\{ {{B_{{\rm{1}}k}}, {B_{{\rm{2}}k}}, {B_{3k}}} \right\} \in {{\bf{R}}^{{{n}} \times {{n}}}} $ is constant matrix.

    We define the FPLS error as

    $ \begin{align} \left\{ \begin{array}{l} {e_1}(t) = {\alpha _1}(t){x_1}(t - r) - {y_1}(t) \\ {e_2}(t) = {\alpha _2}(t){x_2}(t - r) - {y_2}(t) \\ {e_3}(t) = {\alpha _3}(t){x_3}(t - r) - {y_3}(t). \\ \end{array} \right.\\[-5mm] \end{align} $

    (21)

    Based on Theorem 1, the adaptive-impulsive controller is constructed as

    $ \begin{align} \left\{ \begin{array}{l} {u_1} = {{\dot \alpha }_1}(t){x_1}(t - r) - {l_1}{e_1} \\ {u_2} = {{\dot \alpha }_2}(t){x_2}(t - r) - {l_2}{e_2} - \\ \; \; \qquad {\alpha _2}(t){x_1}(t - r){x_3}(t - r) + {y_1}{y_3}\\ {u_3} = {{\dot \alpha }_3}(t){x_3}(t - r) - {l_3}{e_3} + \\ \; \; \qquad {\alpha _3}(t){x_1}(t - r){x_2}(t - r) - {y_1}{y_2} \\ \end{array} \right.\\[-5mm] \end{align} $

    (22)

    where $ l_1 $, $ l_2 $ and $ l_3 $ are constants to be determined. Then, the response system and the error system are described as

    $ \begin{align} \left\{ \begin{array}{l} {{\dot y}_1} = a({y_2} - {y_1}) + {{\dot \alpha }_1}(t){x_1}(t - r) - {l_1}{e_1}, \; \; t \ne {t_k}\\ \Delta {y_1} = {B_{1k}}({\alpha _1}(t){x_1}(t - r) - {y_1}), \; \; t = {t_k} \\ {{\dot y}_2} = b{y_1} + c{y_2} + {{\dot \alpha }_2}(t){x_2}(t - r) - \\\; \; \qquad {\alpha _2}(t){x_1}(t - r){x_3}(t - r) - {l_2}{e_2}, \; \; t \ne {t_k} \\ \Delta {y_2} = {B_{2k}}({\alpha _2}(t){x_2}(t - r) - {y_2}), \; \; t = {t_k}\\ {{\dot y}_3} = - d{y_3} + {{\dot \alpha }_3}(t){x_3}(t - r) + \\\; \; \qquad {\alpha _3}(t){x_1}(t - r){x_2}(t - r) - {l_3}{e_3}, \; \; t \ne {t_k}\\ \Delta {y_3} = {B_{3k}}({\alpha _3}(t){x_3}(t - r) - {y_3}), \; \; t = {t_k} \\\; \; \qquad k = 1, 2, \cdots \end{array} \right.\\[-5mm] \end{align} $

    (23)

    $ \begin{align} \left\{ \begin{array}{l} {{\dot e}_1} = a({e_2} - {e_1}) + {l_1}{e_1}, \; \; t \ne {t_k} \\ \Delta {e_1} = - {B_{1k}}({\alpha _1}(t){x_1}(t - r) - {y_1}), \; \; t = {t_k} \\ {{\dot e}_2} = b{e_1} + c{e_2} + {l_2}{e_2}, \; \; t \ne {t_k}\\ \Delta {e_2} = - {B_{2k}}({\alpha _2}(t){x_2}(t - r) - {y_2}), \; \; t = {t_k}\\ {{\dot e}_3} = - d{e_3} + {l_3}{e_3}, \; \; t \ne {t_k}\\ \Delta {e_3} = - {B_{3k}}({\alpha _3}(t){x_3}(t - r) - {y_3}), \; \; t = {t_k} \\\; \; \qquad k = 1, 2, \cdots \end{array}\right.\\[-5mm] \end{align} $

    (24)
  • Numerical simulation is performed. The solver DDE23 in Matlab is used to integrate the differential equations. The system parameters are chosen as $ a = 35, b = -7, c = 20, d = 3 $, such that the drive system (19) and the response system (20) can exhibit chaotic behaviors without control. The initial conditions of the drive system (19) and the response system (20) are taken as $ {\left({{x_{\rm{1}}}\left(0 \right), {x_{\rm{2}}}\left(0 \right), {x_{\rm{3}}}\left(0 \right)} \right)^{\rm T}} = {\left({{\rm{4}}, {\rm{ 6}}, {\rm{ 8}}} \right)^{\rm T}} $ and $ {\left({{y_{\rm{1}}}\left(0 \right), {y_{\rm{2}}}\left(0 \right), {y_{\rm{3}}}\left(0 \right)} \right)^{\rm T}} = {\left({{\rm{1}}, {\rm{ 2}}, {\rm{ 24}}} \right)^{\rm T}} $, respectively. The initial value of the error system is selected as $ {\left({{e_{\rm{1}}}\left(0 \right), {e_{\rm{2}}}\left(0 \right), {e_{\rm{3}}}\left(0 \right)} \right)^{\rm T}} = {\left({{\rm{3}}, {\rm{ 3}}, {\rm{ 3}}} \right)^{\rm T}} $. The time delay is set as $ r = 0.01 $s, and other controlling parameters are $ {{B_k}} = {\rm{diag}}\left\{ {0.{\rm{999}}, {\rm{ }}0.{\rm{999}}, {\rm{ }}0.{\rm{999}}} \right\} $, $ {{l}} = {\rm{diag}}\left\{ { - {\rm{14}}, {\rm{ }} - {\rm{25}}, {\rm{ }} - {\rm{3}}} \right\} $, impulsive interval $ {t_k} - {t_k}_{ - {\rm{1}}} $ = 0.04 s.

    Firstly, we select the scaling function matrix as $ {\alpha } \left(t \right) = {\rm{diag}}\left\{ {{\rm{1}}, {\rm{ 1}}, {\rm{ 1}}} \right\}, $ i.e., lag synchronization. The corresponding simulation results are illustrated in Figs. 2 and 3. Fig. 2 shows the lag synchronization state variables with the time delay $ r = 0.01 $ s. Fig. 3 displays that the synchronization error variable $ e_1 $, $ e_2 $ and $ e_3 $ converge to zero after a transient time, respectively. From Figs. 2 and 3, one can see that all the state variables of the chaotic system achieve lag synchronization, which shows the correctness and effectiveness of our method via adaptive-impulsive control.

    Figure 2.  Function projective lag synchronization with r = 0.01 s and $ {\alpha } \left(t \right) = {\rm{diag}}\{ 1, 1, 1\} $

    Figure 3.  Time evolution of system errors between systems (19) and (20)

    Secondly, when the scaling function matrix is that $ {\alpha } \left(t \right) = {\rm{diag}}\left\{ {{\cos(t)}, {\cos(t)}, {\cos(t)}} \right\} $, and it means function projective lag synchronization. Fig. 4 displays the time evolution of the state variables of the drive-response system. Fig. 5 depicts the synchronization errors fast converge to zero. These results show that FPLS of systems (19) and (20) has been achieved via adaptive-impulsive control.

    Figure 4.  Function projective lag synchronization with $ r = 0.01 $s and $ {\alpha } \left(t \right) = {\rm{diag}}\left\{ {{{\cos(t)}}, {{\cos(t)}}, {{\cos(t)}}} \right\} $

    Figure 5.  Time evolution of system errors between systems (19) and (20)

    Finally, when time delay $ r = 0 $, function projective lag synchronization becomes function projective synchronization, Fig. 6 shows the system errors converge to zero, and the synchronization is attained. The result agrees with that of Wu and Cao[35], and our scheme is more universally applicable.

    Figure 6.  Time evolution of system errors between systems (19) and (20) when $ r = 0 $

  • In this section, another example is presented to verify the effectiveness of our method.

  • Then we consider the 4D Qi hyperchaotic system[40]:

    $ \begin{align} \left\{ \begin{array}{l} {{\dot x}_1} = a({x_2} - {x_1}) + {x_2}{x_3}{x_4} \\ {{\dot x}_2} = b({x_1} + {x_2}) - {x_1}{x_3}{x_4} \\ {{\dot x}_3} = - c{x_3} + {x_1}{x_2}{x_4} \\ {{\dot x}_4} = - d{x_4} + {x_1}{x_2}{x_3} \\ \end{array} \right.\\[-5mm] \end{align} $

    (25)

    where $ x_1 $, $ x_2 $, $ x_3 $ and $ x_4 $ are state variables, and a, b, c and d are system parameters. When $ c = {c_{\rm{1}}} + {c_{\rm{2}}}{\rm{sin}}\left(t \right) $ and a = 35, b = 10, $ c_1 = 95 $, $ c_2 = 10 $ and d = 10, the system is hyperchaotic. The dynamics of this system have been extensively investigated by Qi et al.[40]

    The response system via adaptive-impulsive control is described as

    $ \begin{align} \left\{ \begin{array}{l} {{\dot y}_1} = a({y_2} - {y_1}) + {u_1}, \; \; t \ne {t_k}\\ {{\dot y}_2} = b{y_1} + c{y_2} - {y_1}{y_3} + {u_2}, \; \; t \ne {t_k}\\ {{\dot y}_3} = - d{y_3} + {y_1}{y_2} + {u_3}, \; \; t \ne {t_k}\\ {{\dot y}_4} = - d{y_4} + {y_1}{y_2}{y_3} + {u_4}, \; \; t \ne {t_k}\\ \Delta {{y}} = y(t_k^ + ) - y({t_k}) = \\\; \; \qquad {{{B_k}}}({{\alpha }}(t){{x}}(t - r) - {{y}}), \; \; t = {t_k}\\\; \; \qquad k = 1, 2, \cdots \end{array} \right.\\ \end{align} $

    (26)

    where $ {{y}} = (y_1, y_2, y_3, y_4)^{\rm T} \in {{\bf{R}}_{{n}}} $ is state variable, $ {\alpha } \left(t \right) = {\rm{diag}}\left({{\alpha _{\rm{1}}}\left(t \right), {\alpha _{\rm{2}}}\left(t \right), {\alpha _{\rm{3}}}\left(t \right), {\alpha _{\rm{4}}}\left(t \right)} \right) $ is scaling function matrix, r is time delay, $ {{B_k}} = {\rm{diag}}\left\{ {{B_{{\rm{1}}k}}, {B_{{\rm{2}}k}}, {B_{{\rm{3}}k}}, , {B_{{\rm{4}}k}}, } \right\} \in {{\bf{R}}^{{{n}} \times {{n}}}} $ are constant matrices.

    We define the FPLS error as

    $ \begin{align} \left\{ \begin{array}{l} {e_1}(t) = {\alpha _1}(t){x_1}(t - r) - {y_1}(t) \\ {e_2}(t) = {\alpha _2}(t){x_2}(t - r) - {y_2}(t) \\ {e_3}(t) = {\alpha _3}(t){x_3}(t - r) - {y_3}(t) \\ {e_4}(t) = {\alpha _4}(t){x_4}(t - r) - {y_4}(t). \end{array} \right.\\[-5mm] \end{align} $

    (27)

    In order to achieve function projective lag synchronization, we construct the following controller:

    $ \begin{align} \left\{ \begin{array}{l} {u_1} = {{\dot \alpha }_1}(t){x_1}(t - r) + \\\; \; \qquad {\alpha _1}(t){x_2}(t - r){x_3}(t - r){x_4}(t - r) - {y_2}{y_3}{y_4} - {l_1}{e_1}\\ {u_2} = {{\dot \alpha }_2}(t){x_2}(t - r) - \\\; \; \qquad {\alpha _2}(t){x_1}(t - r){x_3}(t - r){x_4}(t - r) + {y_1}{y_3}{y_4} - {l_2}{e_2}\\ {u_3} = {{\dot \alpha }_3}(t){x_3}(t - r) + \\\; \; \qquad {\alpha _3}(t){x_1}(t - r){x_2}(t - r){x_4}(t - r) - {y_1}{y_2}{y_4} - {l_3}{e_3}\\ {u_4} = {{\dot \alpha }_4}(t){x_4}(t - r) + \\\; \; \qquad {\alpha _4}(t){x_1}(t - r){x_2}(t - r){x_3}(t - r) - \\\; \; \qquad {y_1}{y_2}{y_3} - {l_4}{e_4} \end{array} \right.\\[-5mm] \end{align} $

    (28)

    where $ l_1 $, $ l_2 $, $ l_3 $ and $ l_4 $ are constants to be determined. Then the response system is described by

    $ \begin{align} \left\{ \begin{array}{l} {{\dot y}_1} = a({y_2} - {y_1}) + {{\dot \alpha }_1}(t){x_1}(t - r) + \\\; \; \qquad {\alpha _1}(t){x_2}(t - r){x_3}(t - r){x_4}(t - r) - {l_1}{e_1}, \; \; t \ne {t_k}\\ \Delta {y_1} = {B_{1k}}({\alpha _1}(t){x_1}(t - r) - {y_1}), \; \; t = {t_k}\\ {{\dot y}_2} = b{y_1} + c{y_2} + {{\dot \alpha }_2}(t){x_2}(t - r) - \\\; \; \qquad {\alpha _2}(t){x_1}(t - r){x_3}(t - r){x_4}(t - r) - {l_2}{e_2}, \; \; t \ne {t_k}\\ \Delta {y_2} = {B_{2k}}({\alpha _2}(t){x_2}(t - r) - {y_2}), \; \; t = {t_k}\\ {{\dot y}_3} = - c{y_3} + {{\dot \alpha }_3}(t){x_3}(t - r) + \\\; \; \qquad {\alpha _3}(t){x_1}(t - r){x_2}(t - r){x_4}(t - r) - {l_3}{e_3}, \; \; t \ne {t_k}\\ \Delta {y_3} = {B_{3k}}({\alpha _3}(t){x_3}(t - r) - {y_3}), \; \; t = {t_k}\\ {{\dot y}_4} = - d{y_3} + {{\dot \alpha }_4}(t){x_4}(t - r) + \\\; \; \qquad {\alpha _4}(t){x_1}(t - r){x_2}(t - r){x_3}(t - r) - {l_4}{e_4}, \; \; t \ne {t_k}\\ \Delta {y_4} = {B_{4k}}({\alpha _4}(t){x_4}(t - r) - {y_4}), \; \; t = {t_k}\\\; \; \qquad k = 1, 2, \cdots \end{array}\right.\\[-5mm] \end{align} $

    (29)

    The error system is given as

    $ \begin{align} \left\{ \begin{array}{l} {{\dot e}_1} = a({e_2} - {e_1}) + {l_1}{e_1}, \; \; t \ne {t_k}\\ \Delta {e_1} = - {B_{1k}}({\alpha _1}(t){x_1}(t - r) - {y_1}), \; \; t = {t_k}\\ {{\dot e}_2} = b{e_1} + b{e_2} + {l_2}{e_2}, \; \; t \ne {t_k}\\ \Delta {e_2} = - {B_{2k}}({\alpha _2}(t){x_2}(t - r) - {y_2}), \; \; t = {t_k}\\ {{\dot e}_3} = - c{e_3} + {l_3}{e_3}, \; \; t \ne {t_k}\\ \Delta {e_3} = - {B_{3k}}({\alpha _3}(t){x_3}(t - r) - {y_3}), \; \; t = {t_k}\\ {{\dot e}_4} = - d{e_4} + {l_4}{e_4}, \; \; t \ne {t_k}\\ \Delta {e_3} = - {B_{4k}}({\alpha _4}(t){x_4}(t - r) - {y_4}), \; \; t = {t_k}\\\; \; \qquad k = 1, 2, \cdots \end{array} \right.\\[-5mm] \end{align} $

    (30)
  • In the numerical simulation, the system parameters are selected as $ a = 35 $, $ b = 10 $, $ c_1 = 95 $, $ c_2 = 10 $ and $ d = 10 $ such that the drive system (25) and the response system (26) are hyperchaotic with no control applied. The initial conditions of the systems (25) and (26) are taken as $ {{x}}\left(0 \right) = {\left({{\rm{1}}, {\rm{ 2}}, {\rm{ 3}}, {\rm{ 4}}} \right)^{\rm T}}, $ $ {{y}}\left(0 \right) = {\left({{\rm{3}}, {\rm{ 5}}, {\rm{ 5}}, {\rm{ 6}}} \right)^{\rm T}} $, respectively. The control parameters are taken as $ {{{B_k}}} = {\rm{diag}}\left\{ {0.{\rm{999}}, {\rm{ }}0.{\rm{999}}, {\rm{ }}0.{\rm{999}}, {\rm{ }}0.{\rm{999}}}\right\} $, impulsive interval $ {t_k} - {t_k}_{ - {\rm{1}}} $ = 0.04 s, time delay $ r = 0.01 $s and $ { {{l}}} = {\rm{diag}}\left\{ { - {\rm{14}}, {\rm{ }} - {\rm{25}}, {\rm{ }} - {\rm{3}}, {\rm{ }} - {\rm{3}}} \right\} $, scaling function matrix is $ {\alpha } \left(t \right) = {\rm{diag}}\left\{ {{{\cos(t)}}, {{\cos(t)}}, {{\cos(t)}}} \right\}. $ Fig. 7 illustrates the state variables and Fig. 8 depicts the synchronization errors. From Figs. 7 and 8, we can see that the function projective lag synchronization between the drive system (25) and the response system (26) is realized by adaptive-impulsive control and our method is effective and correct.

    Figure 7.  Function projective lag synchronization with $ {\alpha } \left(t \right) = {\rm{diag}}\left\{ {{{\cos(t)}}, {{\cos(t)}}, {{\cos(t)}}, {{\cos(t)}}} \right\} $ and r = 0.01 s

    Figure 8.  Time evolution of system errors between systems (25) and (26)

  • In this section, another example is shown to test the effectiveness and generality of our method. In 2002, the unified chaotic system was introduced[41], and it can be presented as follows:

    $ \begin{align} \left\{ \begin{array}{l} {{\dot x}_1} = (25 + a)({x_2} - {x_1})\\ {{\dot x}_2} = (28 - 35a){x_1} - {x_1}{x_3} + (29a - 1){x_2}\\ {{\dot x}_3} = {x_1}{x_2} - \dfrac{{(a + 8){x_3}}}{3} \end{array} \right.\\[-5mm] \end{align} $

    (31)

    where $ x_1 $, $ x_2 $ and $ x_3 $ are state variables, and a is a system parameter, $ a \in \left[{0, 1} \right] $. When $ a \in [0, 0.8) $, the system is Lorenz chaotic system, when $ a = 0.8 $, the system is Lu system, and when $ a \in (0.8, 1] $, the system is Chen system.

    The response system via adaptive-impulsive controller is given as

    $ \begin{align} \quad\left\{ \begin{array}{l} {{\dot y}_1} = (25 + a)({y_2} - {y_1}) + {u_1}, \; \; t \ne {t_k}\\ {{\dot y}_2} = (28 - 35a){y_1} - {y_1}{y_3} + (29a - 1){y_2} + {u_2}, \; \; t \ne {t_k}\\ {{\dot y}_3} = {y_1}{y_2} - \dfrac{{(a + 8){y_3}}}{3} + {u_3}, \; \; t \ne {t_k}\\ \Delta {{y}} = y(t_k^ + ) - y({t_k}) = \\ \; \; \qquad{{{B_k}}}({{\alpha }}(t){{x}}(t - r) - {{y}}), \; \; t = {t_k} \end{array} \right.\\[-5mm] \end{align} $

    (32)

    where $ {{y}} = (y_1, y_2, y_3)^{\rm T} \in {{\bf{R}}_{{n}}} $ is state variables; $ {\alpha } \left(t \right) = {\rm{diag}}\left({{\alpha _{\rm{1}}}\left(t \right), {\alpha _{\rm{2}}}\left(t \right), {\alpha _{\rm{3}}}\left(t \right)} \right) $ is scaling function matrix, r is time delay, $ {{B_k}} = {\rm{diag}}\left\{ {{B_{{\rm{1}}k}}, {B_{{\rm{2}}k}}, {B_{3k}}} \right\} \in {{\bf{R}}^{{{n}} \times {{n}}}} $ is constant matrix.

    The FPLS error is defined as follows:

    $ \begin{align} \left\{ \begin{array}{l} {e_1}(t) = {\alpha _1}(t){x_1}(t - r) - {y_1}(t)\\ {e_2}(t) = {\alpha _2}(t){x_2}(t - r) - {y_2}(t)\\ {e_3}(t) = {\alpha _3}(t){x_3}(t - r) - {y_3}(t). \end{array} \right.\\[-5mm] \end{align} $

    (33)

    Based on {Theorem 1}, the adaptive-impulsive controller is designed as

    $ \begin{align} \left\{ \begin{array}{l} {u_1} = {{\dot \alpha }_1}(t){x_1}(t - r) - {l_1}{e_1}\\ {u_2} = {{\dot \alpha }_2}(t){x_2}(t - r) - {l_2}{e_2} - \\\; \; \qquad {\alpha _2}(t){x_1}(t - r){x_3}(t - r) + {y_1}{y_3}\\ {u_3} = {{\dot \alpha }_3}(t){x_3}(t - r) - {l_3}{e_3} + \\\; \; \qquad {\alpha _3}(t){x_1}(t - r){x_2}(t - r) - {y_1}{y_2} \end{array} \right.\\[-6mm] \end{align} $

    (34)

    where $ l_1 $, $ l_2 $ and $ l_3 $ are constants to be determined. Then, the response system and the error dynamical system are characterized by

    $ \begin{align} \quad\left\{ \begin{array}{l} {{\dot y}_1} = (25 + a)({y_2} - {y_1}) + {{\dot \alpha }_1}(t){x_1}(t - r) - {l_1}{e_1}, \; \; t \ne {t_k}\\ \Delta {y_1} = {B_{1k}}({\alpha _1}(t){x_1}(t - r) - {y_1}), \; \; t = {t_k}\\ {{\dot y}_2} = (28 - 35a){y_1} + (29a - 1){y_2} +{{\dot \alpha }_2}(t){x_2}(t - r) - \\\; \; \qquad {\alpha _2}(t){x_1}(t - r){x_3}(t - r) - {l_2}{e_2}, \; \; t \ne {t_k}\\ \Delta {y_2} = {B_{2k}}({\alpha _2}(t){x_2}(t - r) - {y_2}), \; \; t = {t_k}\\ {{\dot y}_3} = - \dfrac{{(a + 8){y_3}}}{3} + {{\dot \alpha }_3}(t){x_3}(t - r) + \\\; \; \qquad {\alpha _3}(t){x_1}(t - r){x_2}(t - r) - {l_3}{e_3}, \; \; t \ne {t_k}\\ \Delta {y_3} = {B_{3k}}({\alpha _3}(t){x_3}(t - r) - {y_3}), \; \; t = {t_k}\\\; \; \qquad k = 1, 2, \cdots \end{array} \right.\\ \end{align} $

    (35)

    $ \begin{align} \left\{ \begin{array}{l} {{\dot e}_1} = (25 + a)({e_2} - {e_1}) + {l_1}{e_1}, \; \; t \ne {t_k}\\ \Delta {e_1} = - {B_{1k}}({\alpha _1}(t){x_1}(t - r) - {y_1}), \; \; t = {t_k}\\ {{\dot e}_2} = (28 - 35a){e_1} + (29a - 1){e_2} + {l_2}{e_2}, \; \; t \ne {t_k}\\ \Delta {e_2} = - {B_{2k}}({\alpha _2}(t){x_2}(t - r) - {y_2}), \; \; t = {t_k}\\ {{\dot e}_3} = - \dfrac{{(a + 8){e_3}}}{3} + {l_3}{e_3}, \; \; t \ne {t_k}\\ \Delta {e_3} = - {B_{3k}}({\alpha _3}(t){x_3}(t - r) - {y_3}), \; \; t = {t_k}\\\; \; \qquad k = 1, 2, \cdots \end{array} \right.\\[-5mm] \end{align} $

    (36)

    Numerical simulation is performed. The solver DDE23 in Matlab is used to solve the differential equations. The initial conditions of the drive system (31) and the response system (32) are randomly taken as $ {\left({{x_{\rm{1}}}\left(0 \right), {x_{\rm{2}}}\left(0 \right), {x_{\rm{3}}}\left(0 \right)} \right)^{\rm T}} = {\left({{\rm{4}}, {\rm{ 6}}, {\rm{ 8}}} \right)^{\rm{T}}} $ and $ {\left({{y_{\rm{1}}}\left(0 \right), {y_{\rm{2}}}\left(0 \right), {y_{\rm{3}}}\left(0 \right)} \right)^{\rm T}} = {\left({{\rm{1}}, {\rm{ 2}}, {\rm{ 24}}} \right)^{\rm T}} $, respectively. The initial value of the error system is selected as $ {\left({{e_{\rm{1}}}\left(0 \right), {e_{\rm{2}}}\left(0 \right), {e_{\rm{3}}}\left(0 \right)} \right)^{\rm T}} = {\left({{\rm{3}}, {\rm{ 3}}, {\rm{ 3}}} \right)^{\rm T}} $. The time delay is set as r = 0.01 s, and other controlling parameters are $ {{B_k}} = {\rm{diag}}\left\{ {0.{\rm{999}}, {\rm{ }}0.{\rm{999}}, {\rm{ }}0.{\rm{999}}} \right\} $, $ {{l}} = {\rm{diag}}\left\{ { - {\rm{14}}, {\rm{ }} - {\rm{25}}, {\rm{ }} - {\rm{3}}} \right\} $, impulsive interval $ {t_k} - {t_k}_{ - {\rm{1}}} $ = 0.04 s. The scaling function matrix is that $ {\alpha } \left(t \right) = {\rm{diag}}\left\{ {{{\cos(t)}}, {{\cos(t)}}, {\rm{ cos(t)}}} \right\} $, and it means function projective lag synchronization. The numerical simulation results are shown in Figs. 9 and 10 ($ a = 0.8 $), Figs. 11 and 12 ($ a = 0.9 $). In Figs. 9 and 10, $ a = 0.8 $, and the system is Lu system. In Figs. 11 and 12, $ a = 0.9 $, and the system is Chen system. From the figures, we can see that the synchronization error variables converge to zero after some time, and the function projective lag synchronization between the drive system and the response system have been attained.

    Figure 9.  Function projective lag synchronization with $ {\alpha } \left(t \right) = {\rm{diag}}\left\{ {{{\cos(t)}}, {{\cos(t)}}, {{\cos(t)}}, {{\cos(t)}}} \right\} $ and $ r = 0.01 $s, $ a = 0.8 $

    Figure 10.  Time evolution of system errors ($a = 0.8$)

    Figure 11.  Function projective lag synchronization with $ {\alpha } \left(t \right) = {\rm{diag}}\left\{ {{{\cos(t)}}, {{\cos(t)}}, {{\cos(t)}}, {{\cos(t)}}} \right\} $ and r = 0.01 s, $ a = 0.9 $

    Figure 12.  Time evolution of system errors ($a = 0.9$)

  • In this paper, we study the function projective lag synchronization of chaotic and hyperchaotic systems using the adaptive-impulsive control technology for the first time. Based on the Lyapunov stability theory and the impulsive control technology, suitable nonlinear adaptive-impulsive controllers are designed, some effective and practical sufficient conditions are presented, and function projective lag synchronization has been realized. The controller is designed simply and efficiently.

    Numerical simulations show the effectiveness and correctness of our method. Our designed controllers have potential applications in secure communication, life science, detuned laser, rotating fluids, waves and turbulence in chemical oscillations and information engineering. It is worth noting that the complete synchronization, projective synchronization, function projective synchronization, lag synchronization are all belong to FPLS. Thus, our obtained results are applicable and representative, and they can be applied to many chaotic systems.

    In practical situations, the parameters of the drive and response systems are sometimes unknown, probably there are delays in the drive and response systems, and sometimes the chaotic systems are fractional-order systems, then our results may be inapplicable, so in the following work, we will continue to improve our controllers and make them efficiently used in the actual application.

  • This work was supported by National Natural Science Foundation of China (Nos. 41571417 and U1604145), Science and Technology Foundation of Henan Province of China (No. 152102210048), Foundation and Frontier Project of Henan Province of China (No. 162300410196), China Postdoctoral Science Foundation (No. 2016M602235), Natural Science Foundation of Educational Committee of Henan Province of China (No. 14A413015), and Research Foundation of Henan University (No. xxjc20140006).

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