^{1}

^{*}

^{2}

^{3}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this paper, we investigate adaptive switched generalized function projective synchronization between two new different hyperchaotic systems with unknown parameters, which is an extension of the switched modified function projective synchronization scheme. Based on the Lyapunov stability theory, corresponding adaptive controllers with appropriate parameter update laws are constructed to achieve adaptive switched generalized function projective synchronization between two different hyperchaotic systems. A numerical simulation is conducted to illustrate the validity and feasibility of the proposed synchronization scheme.

Hyperchaos, which was first introduced by Rössler [

Since the concept of synchronizing two identical chaotic systems from different initial conditions was introduced by Pecora and Carroll in 1990 [

Among the above-mentioned synchronization phenomena, projective synchronization has been investigated with increasing interest in recent years due to the fact that it can obtain faster communication with its proportional feature [

Inspired by the previous works, in this paper, we propose the switched generalized function projective synchronization (SGFPS) between two different hyperchaotic systems using adaptive control method by extending the GFPS and SMFPS schemes, in which a state variable of the drive system synchronizes with a different state variable of the response system up to a more generalized scaling function matrix. Due to the unpredictability of the switched states and scaling function matrix, this synchronization scheme can provide additional security in secure communication.

The rest of this paper is organized as follows. Section 2 gives a brief description of the SGFPS scheme and two new hyperchaotic systems. In Section 3, we propose appropriate adaptive controllers and parameter update laws for the adaptive switched generalized function projective synchronization of two different hyperchaotic systems. Section 4 presents a numerical example to illustrate the effectiveness of the proposed method. Finally, conclusions are given in Section 5.

Consider the following drive and response systems:
^{n}^{n}^{n}

The error states between the drive and response systems are defined as
_{i}^{n}_{1}(_{2}(_{n}

Recently, Li ^{T}

Lately, Dadras ^{T}

For more information on the dynamical behaviors of these two systems, please refer to [

In this section, we investigate the adaptive SGFPS between systems

Suppose that system _{1}, _{1}, _{1}, _{1}, _{1}, _{1}, _{2}, _{2} and _{2} are unknown parameters to be identified, and _{i}_{i}

So the SGFPS error dynamical system is determined as follows

Without loss of generality, the scaling functions can be chosen as _{1}(_{11}_{1} + _{12}, _{2}(_{21}_{1} + _{22}, _{3}(_{31}_{1} + _{32} and _{4}(_{41}_{1} + _{42}, where _{ij}

Our goal is to find the appropriate controllers _{i}_{1}, _{2}, _{3}, _{4}) is a positive gain matrix for each state controller. In practical applications the synchronization process can be sped up by increasing the gain matrix

The update laws for the unknown parameters _{1}, _{1}, _{1}, _{1}, _{1}, _{1}, _{2}, _{2} and _{2} are given as follows
_{1}, _{1}, _{1}, _{1}, _{1}, _{1}, _{2}, _{2} and _{2} are the estimate values for these unknown parameters, respectively. Then, we have the following main result.

_{1}(_{2}(_{3}(_{4}(

Substituting

Since the Lyapunov function

This completes the proof.

In this section, to verify and demonstrate the effectiveness of the proposed method we consider a numerical example. In the numerical simulations, the fourth-order Runge-Kutta method is used to solve the systems with time step size 0.001. The true values of the “unknown” parameters of systems _{1} = 12, _{1} = 23, _{1} = 1, _{1} = 2.1, _{1} = 6, _{1} = 0.2, _{2} = 8, _{2} = 40, _{2} = 14.9, so that the two systems exhibit hyperchaotic behavior, respectively. The initial values for the drive and response systems are _{1}(0) = 8.3, _{1}(0) = 10.8, _{1}(0) = 17.4, _{1}(0) = −11.1, _{2}(0) = −0.2, _{2}(0) = −0.1, _{2}(0) = 16.9 and _{2}(0) = −0.7, and the estimated parameters have initial conditions 0.1. Given that the function factors are _{1}(_{1} − 0.3, _{2}(_{1} +0.5, _{3}(_{1} +0.03, _{4}(_{1} + 0.03, and the gain matrix

In this paper, we have investigated switched generalized function projective synchronization between two new different hyperchaotic systems with fully unknown parameters, which extended the switched modified function projective synchronization scheme. In this synchronization scheme, a state variable of the drive system synchronizes with a different state variable of the response system up to a generalized scaling function matrix. Due to the unpredictability of the switched states and scaling function matrix, this synchronization scheme can provide additional security in secure communication. By applying the adaptive control theory and Lyapunov stability theory, the appropriate adaptive controllers with parameter update laws are proposed to achieve SGFPS between two different hyperchaotic systems. A numerical simulation was conducted to illustrate the validity and feasibility of the proposed synchronization scheme.

This work was supported by the Youth Foundation of Yunnan University of Nationalities under grant No.11QN07, the Natural Science Foundation of Yunnan Province under grants No.2009CD019 and No.2011FZ172, the Natural Science Foundation of China under grant No.61263042.

The authors declare no conflict of interest.

Hyperchaotic attractor of system

Hyperchaotic attractor of system

The time evolution of SGFPS errors for the drive system _{1} = _{2}−(2_{1} − 0.3)_{1}, _{2} = _{2} − (2_{1} + 0.5)_{1}, _{3} = _{2} − (0.5_{1} + 0.03)_{1}, _{4} = _{2} − (−0.5_{1} + 0.03)_{1}.

The time evolution of the estimated unknown parameters of system

The time evolution of the estimated unknown parameters of system