Adaptive Modified Function Projective Synchronization between Two Different Hyperchaotic Dynamical Systems

Abstract

This work investigates modified function projective synchronization between two different hyperchaotic dynamical systems, namely, hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between two diffierent hyperchaotic dynamical systems. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.

1. Introduction

During the last three decades, synchronization of chaotic systems has attracted increasing attention from scientists and engineers and has been explored intensively both theoretically and experimentally. Since Pecora and Carrol [1] introduced a method to synchronize two identical systems with different initial conditions, many approaches have been proposed for the synchronization of chaotic or hyperchaotic systems such as complete synchronization [1], phase synchronization [2], generalized synchronization [3], lag synchronization [4], intermittent lag synchronization [5], time-scale synchronization [6], intermittent generalized synchronization [7], projective synchronization [8], modified projective synchronization [9, 10], and function projective synchronization [11, 12]. Most of them are based on exactly knowing the system structure and parameters, but in practice, some or all of the system's parameters are unknown. Moreover, these parameters change from time to time. A lot of works have been done to solve this problem using adaptive synchronization [13–16]. Most of the methods mentioned above synchronize two identical chaotic systems. Hyperchaotic system is usually classified as a chaotic system with more than one positive Lyapunov exponent, indicating that the chaotic dynamics of the system are expanded in more than one direction giving rise to a more complex attractor. The method of the synchronization of two different hyperchaotic systems is far from being straightforward. There is little work about this challenging problem because it consists of different structures and parameter mismatch of the two hyperchaotic systems. Complete synchronization is characterized by the equality of state variables while evolving in time. Antisynchronization is characterized by the vanishing of the sum of relevant variables. Projective synchronization occurs when the drive and response system could be synchronized up to a scaling factor. Function projective synchronization is the most general definition of projective synchronization. It means that the derive and response systems could be synchronized up to a scaling function.

A focused problem in the study of chaos synchronization is how to design a physically available and simple controller to guarantee the realization of high-quality synchronization in coupled chaotic systems. Linear feedback is of course a practical technique, but the shortcoming is that it needs to find the suitable feedback constant. Recently, Huang proposed a simple adaptive feedback control method, which neednot to estimate or find feedback constant, to effectively synchronize two almost arbitrary identical chaotic systems in his series paper [17–19].

In this work, we investigate modified function projective synchronization (MFPS) between hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. This work is organized as follows. In Section 2 the modified function projective synchronization (MFPS) scheme is presented. Section 3 briefly describes hyperchaotic Lorenz system and hyperchaotic Chen system. Section 4 proposes adaptive control laws and parameter update rules for the modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Chen system. In Section 5, numerical examples are given to demonstrate the effectiveness of the proposed method. Finally, the conclusions are given in Section 6.

2. Adaptive Modified Function Projective Synchronization (MFPS) Scheme

Consider the following master and slave system: 𝑥̇=𝑓(𝑥,𝑡),𝑦̇=𝑔𝑦,𝑡+𝑢𝑥,𝑦,𝑡, (2.1)(2.2) where 𝑥,𝑦∈𝑅^𝑛 are the state vector of the system (2.1) and (2.2), respectively; 𝑓,𝑔∶𝑅^𝑛→𝑅^𝑛 are two continuous nonlinear vector functions, 𝑢(𝑥,𝑦,𝑡) is the vector controller. We define the error dynamical system as 𝑒(𝑡)=𝑦−𝑀ℎ(𝑡)𝑥, (2.3) where 𝑀 is a constant diagonal matrix 𝑀=diag{𝑚₁,𝑚₂,…,𝑚_𝑛}∈𝑅^𝑛×𝑛 and ℎ(𝑡) a continuous differentiable function with ℎ(𝑡)≠0 for all 𝑡.

The system (2.1) and (2.2) i said to be in modified function projective synchronization if there exists a constant diagonal matrix 𝑀 and function ℎ(𝑡), such that Lim_𝑡→∞‖𝑒(𝑡)‖=0.

It is easy to see that the definition of modified function projective synchronization encompasses function projective synchronization when the scaling matrix 𝑀 equals 𝐼.

3. System Description

The hyperchaotic Lorenz system is described as follows [20–22]: ,̇𝑥̇=𝛼𝑦−𝑥𝑦̇=𝛽𝑥+𝑦−𝑥𝑧−𝑤,𝑧=𝑥𝑦−𝑏𝑧,𝑤̇=𝑟𝑦𝑧, (3.1) where 𝑥, 𝑦, 𝑧, and 𝑤 are state variables and 𝛼, 𝛽, 𝛾, and 𝑟 are real constant parameters. In [21, 22], it has been shown that the system (3.1) has two positive Lyapunov exponents when 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1, the system (3.1) exhibits hyperchaotic behavior, see Figure 1.

Figure 1 

The attractor of hyperchaotic Lorenz dynamical system at 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1 in 𝑥, 𝑦, 𝑧 subspace.

Hyperchaotic Chen system is described as [23, 24] ̇𝑥̇=𝑎𝑦−𝑥+𝑤,𝑦̇=𝑑𝑥+𝑐𝑦−𝑥𝑧,𝑧=𝑥𝑦−𝑏𝑧,𝑤̇=𝑙𝑤+𝑦𝑧, (3.2) where 𝑥, 𝑦, 𝑧 and 𝑤 are state variables and 𝑎,𝑏,𝑐,𝑑 and ℎ are real constant parameters. When 𝑎=35,𝑏=3,𝑐=12,𝑑=7,0.798≤𝑙≤0.9, system (3.2) is periodic, when 𝑎=35,𝑏=3,𝑐=12,𝑑=7,0≤𝑙≤0.085, system (3.2) is chaotic; when 𝑎=35,𝑏=3,𝑐=12,𝑑=7,0.085≤𝑙≤0.798, system (3.2) exhibits hyperchaotic behavior see Figure 2.

Figure 2 

The attractor of hyperchaotic Chen dynamical system at 𝑎=35, 𝑏=3, 𝑐=12, 𝑑=7, and 𝑙=0.5 in 𝑥, 𝑦, 𝑧 subspace.

4. Adaptive MFPS between Hyperchaotic Lorenz System and Chen System

In order to achieve the synchronization behavior between hyperchaotic Lorenz system and hyperchaotic Chen system, we assume that hyperchaotic Lorenz system is the drive system whose four variables are denoted by subscript 1 and hyperchaotic Chen system is the response system whose variables are denoted by subscript 2. The drive and response systems are described by the following equations, respectively,𝑥̇₁=𝛼𝑦₁−𝑥₁,𝑦₁̇=𝛽𝑥₁+𝑦₁−𝑥₁𝑧₁−𝑤₁,𝑧₁̇=𝑥₁𝑦₁−𝛾𝑧₁,𝑤̇₁=𝑟𝑦₁𝑧₁,𝑥̇₂=𝑎𝑦₂−𝑥₂+𝑤₂+𝑢₁,𝑦₂̇=𝑑𝑥₂+𝑐𝑦₂−𝑥₂𝑧₂+𝑢₂,𝑧₂̇=𝑥₂𝑦₂−𝑏𝑧₂+𝑢₃,𝑤̇₂=𝑙𝑤₂+𝑦₂𝑧₂+𝑢₄, (4.1)(4.2) where 𝑈=[𝑢₁,𝑢₂,𝑢₃,𝑢₄]^𝑇 is the nonlinear controller functions which are to be determined later. The term “synchronization,” in general, means that the signal effect upon the synchronized system is very small in comparison with the amplitude of proper oscillations of the system but, nevertheless, it is enough to change the system behavior and to “impose” the rhythm of external influence to it. The two hyperchaotic dynamical systems can be synchronized in the sense that lim_𝑡→∞𝑥₂−𝑚₁ℎ(𝑡)𝑥₁=0,lim_𝑡→∞𝑦₂−𝑚₂ℎ(𝑡)𝑦₁=0,lim_𝑡→∞𝑧₂−𝑚₃ℎ(𝑡)𝑧₁=0,lim_𝑡→∞𝑤₂−𝑚₄ℎ(𝑡)𝑤₁=0, (4.3) where 𝑚_𝑖, (𝑖=1,2,3,4) is the scaling factor and ℎ(𝑡) the scaling function.

We have the following error dynamical system: 𝑒_𝑥̇=𝑥̇₂−𝑚₁ℎ(𝑡)𝑥̇₁−𝑚₁ℎ̇(𝑡)𝑥₁,𝑒_𝑦̇=𝑦₂̇−𝑚₂ℎ(𝑡)𝑦₁̇−𝑚₂ℎ̇(𝑡)𝑦₁,𝑒_𝑧̇=𝑧₂̇−𝑚₃ℎ(𝑡)𝑧₁̇−𝑚₃ℎ̇(𝑡)𝑧₁,𝑒_𝑤̇=𝑤̇₂−𝑚₄ℎ(𝑡)𝑤̇₁−𝑚₄ℎ̇(𝑡)𝑤₁, (4.4) where 𝑒_𝑥=𝑥₂−𝑚₁ℎ(𝑡)𝑥₁, 𝑒_𝑦=𝑦₂−𝑚₁ℎ(𝑡)𝑦₁, 𝑒_𝑧=𝑧₂−𝑚₁ℎ(𝑡)𝑧₁, and 𝑒_𝑤=𝑤₂−𝑚₁ℎ(𝑡)𝑤₁.

Substitution of (4.1) and (4.2) in (4.4) yields following error dynamical system 𝑒_𝑥̇=𝑎𝑦₂−𝑥₂+𝑤₂+𝑢₁−𝑚₁ℎ(𝑡)𝛼𝑦₁−𝑥₁−𝑚₁ℎ̇(𝑡)𝑥₁,𝑒_𝑦̇=𝑑𝑥₂+𝑐𝑦₂−𝑥₂𝑧₂+𝑢₂−𝑚₂ℎ(𝑡)𝛽𝑥₁+𝑦₁−𝑥₁𝑧₁−𝑤₁−𝑚₂ℎ̇(𝑡)𝑦₁,𝑒_𝑧̇=𝑥₂𝑦₂−𝑏𝑧₂+𝑢₃−𝑚₃ℎ(𝑡)𝑥₁𝑦₁−𝛾𝑧₁−𝑚₃ℎ̇(𝑡)𝑧₁,𝑒_𝑤̇=𝑙𝑤₂+𝑦₂𝑧₂+𝑢₄−𝑚₄ℎ(𝑡)𝑟𝑦₁𝑧₁−𝑚₄ℎ̇(𝑡)𝑤₁. (4.5)

Our aim is to find control laws 𝑢_𝑖(𝑖=1,2,3,4) for stabilizing the error variables of the system at the origin. For this end, we propose following control law:𝑢₁=𝑚₁ℎ(𝑡)𝛼₁𝑦₁−𝑥₁+𝑚₁ℎ̇(𝑡)𝑥₁−𝑎₁𝑥₂−𝑦₂−𝑤₂−𝑘₁𝑒_𝑥,𝑢₂=𝑚₂ℎ(𝑡)𝑦₁+𝛽₁𝑥₁−𝑤₁−𝑥₁𝑧₁+𝑥₂𝑧₂+𝑚₂ℎ̇(𝑡)𝑦₁−𝑑₁𝑥₂−𝑐₁𝑦₂−𝑘₂𝑒_𝑦,𝑢₃=𝑚₃ℎ(𝑡)𝑥₁𝑦₁−𝛾₁𝑧₁+𝑚₃ℎ̇(𝑡)𝑧₁+𝑏₁𝑧₂−𝑥₂𝑦₂−𝑘₃𝑒_𝑧,𝑢₄=𝑚₄ℎ(𝑡)𝑟₁𝑦₁𝑧₁+𝑚₄ℎ̇(𝑡)𝑤₁−𝑙₁𝑤₂−𝑦₂𝑧₂−𝑘₄𝑒_𝑤, (4.6) and the update laws for the unknown parameters 𝛼₁, 𝛽₁, 𝛾₁, 𝑟₁, 𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁, and 𝑙₁ are 𝛼̇₁=𝑥₁−𝑦₁𝑚₁ℎ(𝑡)𝑒_𝑥−𝑘₅𝛼₁,𝛽̇−𝛼₁=−𝑥₁𝑚₂ℎ(𝑡)𝑒_𝑦−𝑘₆𝛽₁−𝛽,𝛾̇₁=𝑧₁𝑚₃ℎ(𝑡)𝑒_𝑧−𝑘₇𝛾₁−𝛾,𝑟₁̇=−𝑦₁𝑧₁𝑚₄ℎ(𝑡)𝑒_𝑤−𝑘₈𝑟₁−𝑟,𝑎₁̇=𝑦₂−𝑥₂𝑒_𝑥−𝑘₉𝑎₁,𝑏̇−𝑎₁=−𝑧₂𝑒_𝑧−𝑘₁₀𝑏₁−𝑏,𝑐₁̇=𝑦₂𝑒_𝑦−𝑘₁₁𝑐₁,𝑑̇−𝑐₁=𝑥₂𝑒_𝑦−𝑘₁₂𝑑₁,𝑙̇−𝑑₁=𝑤₂𝑒_𝑤−𝑘₁₃𝑙₁−𝑙, (4.7) where 𝑘_𝑖>0(𝑖=1,2,3,…,13).

Theorem 4.1. For given constant scaling matrix 𝑀 and scaling function ℎ(𝑡), the MFPS between two systems (4.1) and (4.2) will occur by the control law (4.6) and update law (4.7), and satisfy lim_𝑡→∞𝛼₁−𝛼=lim_𝑡→∞𝛽₁−𝛽=lim_𝑡→∞𝛾₁−𝛾=lim_𝑡→∞𝑟₁−𝑟=lim_𝑡→∞𝑎₁−𝑎=lim_𝑡→∞𝑏₁−𝑏=lim_𝑡→∞𝑐₁−𝑐=lim_𝑡→∞𝑑₁−𝑑=lim_𝑡→∞𝑙₁−𝑙=0. (4.8)

Proof. Define a Lyapunov function, 𝑉(𝑒)=12𝑒²_𝑥+𝑒²_𝑦+𝑒²_𝑧+𝑒²_𝑤+𝑒²_𝛼+𝑒²_𝛽+𝑒²_𝛾+𝑒²_𝑟+𝑒²_𝑎+𝑒²_𝑏+𝑒²_𝑐+𝑒²_𝑑+𝑒²_𝑙, (4.9) where 𝑒_𝛼=𝛼₁−𝛼,𝑒_𝛽=𝛽₁−𝛽,𝑒_𝛾=𝛾₁−𝛾,𝑒_𝑟=𝑟₁−𝑟,𝑒_𝑎=𝑎₁𝑒−𝑎,_𝑏=𝑏₁−𝑏,𝑒_𝑐=𝑐₁−𝑐,𝑒_𝑑=𝑑₁−𝑑,𝑒_𝑙=𝑙₁−𝑙. (4.10)
The time derivative of the Lyapunov function along the trajectory of error system (4.9) is 𝑑𝑉(𝑒)𝑑𝑡=𝑒_𝑥𝑒_𝑥̇+𝑒_𝑦𝑒_𝑦̇+𝑒_𝑧𝑒_𝑧̇+𝑒_𝑤𝑒_𝑤̇+𝑒_𝛼𝑒_𝛼̇+𝑒_𝛽𝑒_𝛽̇+𝑒_𝛾𝑒_𝛾̇+𝑒_𝑟𝑒_𝑟̇+𝑒_𝑎𝑒_𝑎̇+𝑒_𝑏𝑒_𝑏̇+𝑒_𝑐𝑒_𝑐̇+𝑒_𝑑𝑒_𝑑̇+𝑒_𝑙𝑒_𝑙̇=𝑒_𝑥𝑒_𝑥̇+𝑒_𝑦𝑒_𝑦̇+𝑒_𝑧𝑒_𝑧̇+𝑒_𝑤𝑒_𝑤̇+𝑒_𝛼𝛼̇₁+𝑒_𝛽𝛽̇₁+𝑒_𝛾𝛾̇₁+𝑒_𝑟𝑟₁̇+𝑒_𝑎𝑎₁̇+𝑒_𝑏𝑏̇₁+𝑒_𝑐𝑐₁̇+𝑒_𝑑𝑑₁̇+𝑒_𝑙𝑙₁̇. (4.11)
Inserting (4.6) and (4.7) into (4.11) yields the following: 𝑑𝑉(𝑒)𝑑𝑡=−𝑘₁𝑒²_𝑥−𝑘₂𝑒²_𝑦−𝑘₃𝑒²_𝑧−𝑘₄𝑒²_𝑤−𝑘₅𝑒²_𝛼−𝑘₆𝑒²_𝛽−𝑘₇𝑒²_𝛾−𝑘₈𝑒²_𝑟−𝑘₉𝑒²_𝑎−𝑘₁₀𝑒²_𝑏−𝑘₁₁𝑒²_𝑐−𝑘₁₂𝑒²_𝑑−𝑘₁₃𝑒²_𝑙=−𝐾𝑒², (4.12) where 𝑒=(𝑒_𝑥,𝑒_𝑦,𝑒_𝑧,𝑒_𝑤,𝑒_𝛼,𝑒_𝛽,𝑒_𝛾,𝑒_𝑟,𝑒_𝑎,𝑒_𝑏,𝑒_𝑐,𝑒_𝑑,𝑒_𝑙)^𝑇 and 𝐾=diag(𝑘₁,𝑘₂,𝑘₃,𝑘₄,𝑘₅,𝑘₆,𝑘₇,𝑘₈,𝑘₉,𝑘₁₀,𝑘₁₁,𝑘₁₂,𝑘₁₃)^𝑇.
Since 𝑑𝑉(𝑒)/𝑑𝑡≤0, we have 𝑒_𝑥,𝑒_𝑦,𝑒_𝑧,𝑒_𝑤,𝑒_𝛼,𝑒_𝛽,𝑒_𝛾,𝑒_𝑟,𝑒_𝑎,𝑒_𝑏,𝑒_𝑐,𝑒_𝑑,𝑒_𝑙→0 as 𝑡→∞,lim_𝑡→∞‖𝑒‖=0.
Therefore, the drive system (4.1) synchronizes the response system (4.2) in the sense of MFPS.

Remark 4.2. Note that complete synchronization and antisynchronization between two different hyperchaotic dynamical systems are special cases of MFPS with the scaling function ℎ(𝑡)=1 and the scaling factors 𝑚_𝑖=1 and 𝑚_𝑖=−1(𝑖=1,2,3,4), respectively.

Remark 4.3. Note that function projective synchronization (FPS) between two different hyperchaotic dynamical systems is special case of MFPS with the scaling factors 𝑚_𝑖=1(𝑖=1,2,3,4), and the scaling function ℎ(𝑡) is chosen later.

Remark 4.4. Note that generalized projective synchronization (GPS) and modified generalized projective synchronization (MGPS) between two different hyperchaotic dynamical systems are special cases of MFPS with the scaling function ℎ(𝑡)=1 and the scaling factors 𝑚_𝑖 are equal and 𝑚_𝑖 are not equal (𝑖=1,2,3,4), respectively.

By suitable choosing for ℎ(𝑡), we can achieve modified function projective synchronization, complete synchronization, antisynchronization, function projective synchronization, generalized projective synchronization, modified generalized projective synchronization, between two different hyperchaotic systems (see Examples 5.1–5.6).

5. Numerical Results

In this section, numerical examples are used to demonstrate the effectiveness of the proposed method. By using Maple 12 to solve the systems of differential equations (4.1), (4.2), (4.6), and (4.7), we assume that the initial conditions of the drive system are 𝑥₁(0)=2, 𝑦₁(0)=2, 𝑧₁(0)=3, and 𝑤₁(0)=1, and the initial conditions of the response system are 𝑥₂(0)=6, 𝑦₂(0)=5, 𝑧₂(0)=3, and𝑤₂(0)=3. The initial conditions of the estimated parameters are chosen as 𝛼₁(0)=0, 𝛽₁(0)=0, 𝛾₁(0)=0, 𝑟₁(0)=0, 𝑎₁(0)=0, 𝑏₁(0)=0, 𝑐₁(0)=0, 𝑑₁(0)=0 and 𝑙₁(0)=0.

Let the scaling function be ℎ(𝑡)=sin(0.1𝜋𝑡) and the scaling factors are chosen as 𝑚₁=2, 𝑚₂=3, 𝑚₃=5, and 𝑚₄=0.5. The simulation of the error dynamical system between hyperchaotic Lorenz system and hyperchaotic Chen system without control functions is shown in Figure 3(a) displays the 𝑒_𝑥=𝑥₂−𝑚₁ℎ(𝑡)𝑥₁, Figure 3(b) displays the 𝑒_𝑦=𝑦₂−𝑚₁ℎ(𝑡)𝑦₁, Figure 3(c) displays the 𝑒_𝑧=𝑧₂−𝑚₁ℎ(𝑡)𝑧₁, and Figure 3(d) displays the 𝑒_𝑤=𝑤₂−𝑚₁ℎ(𝑡)𝑤₁.

(a)

(b)

(c)

(d)

Figure 3 

The behavior of the trajectories 𝑒𝑥, 𝑒𝑦, 𝑒𝑧, and 𝑒𝑤 of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system without control functions.

Example 5.1. Let the scaling function be ℎ(𝑡)=sin(0.1𝜋𝑡) and the scaling factors are chosen as 𝑚₁=2,𝑚₂=3,𝑚₃=5, and 𝑚₄=0.5. Furthermore, the control gains are chosen as 𝑘₁=𝑘₂=𝑘₃=𝑘₄=3,𝑘₅=𝑘₆=𝑘₇=𝑘₈=𝑘₉=𝑘₁₀=𝑘₁₁=𝑘₁₂=ℎ₁₃=2. Figure 4 displays the MFPS between systems (4.1) and (4.2). Figure 5 show that the estimates 𝛼₁(𝑡), 𝛽₁(𝑡), 𝛾₁(𝑡), 𝑟₁(𝑡) of the unknown parameters converge to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1as 𝑡→∞. Figure 6 show that the estimates 𝑎₁(𝑡), 𝑏₁(𝑡), 𝑐₁(𝑡), 𝑑₁(𝑡), 𝑙₁(𝑡) of the unknown parameters converge to 𝑎=35, 𝑏=3, 𝑐=12, 𝑑=7, and 𝑙=0.5 as 𝑡→∞.

Figure 4 

The behavior of the trajectories 𝑒𝑥, 𝑒𝑦, 𝑒𝑧, and 𝑒𝑤 of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for MFPS.

Figure 5 

The estimates 𝛼1(𝑡), 𝛽1(𝑡), 𝛾1(𝑡), 𝑟1(𝑡) of the unknown parameters converges to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1 as 𝑡 tends to 3.

Figure 6 

The estimates 𝑎1(𝑡), 𝑏1(𝑡), 𝑐1(𝑡), 𝑑1(𝑡), 𝑙1(𝑡) of the unknown parameters converge to 𝑎=35,𝑏=3,𝑐=12,𝑑=7, and 𝑙=0.5 as 𝑡 tends to 3.

Example 5.2. Let the scaling function be ℎ(𝑡)=1 and the scaling factors are chosen as 𝑚₁=𝑚₂=𝑚₃=𝑚₄=1. Furthermore, the control gains are chosen as 𝑘₁=𝑘₂=𝑘₃=𝑘₄=3,𝑘₅=𝑘₆=𝑘₇=𝑘₈=𝑘₉=𝑘₁₀=𝑘₁₁=𝑘₁₂=ℎ₁₃=2. Figure 7(a) displays the complete synchronization between systems (4.1) and (4.2). Figure 7(b) show that the estimates 𝛼₁(𝑡), 𝛽₁(𝑡), 𝛾₁(𝑡), 𝑟₁(𝑡) of the unknown parameters converge to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1as 𝑡→∞. Figure 7(c) show that the estimates 𝑎₁(𝑡), 𝑏₁(𝑡), 𝑐₁(𝑡), 𝑑₁(𝑡), 𝑙₁(𝑡) of the unknown parameters converge to 𝑎=35,𝑏=3,𝑐=12,𝑑=7, and 𝑙=0.5 as 𝑡→∞.

(a)

(b)

(c)

Figure 7 

(a) The behavior of the trajectories 𝑒𝑥,𝑒𝑦,𝑒𝑧,and 𝑒𝑤 of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for complete synchronization. (b) the estimates 𝛼1(𝑡), 𝛽1(𝑡), 𝛾1(𝑡), 𝑟1(𝑡) of the unknown parameters converge to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1as 𝑡 tends to 3. (c) the estimates 𝑎1(𝑡), 𝑏1(𝑡), 𝑐1(𝑡), 𝑑1(𝑡), 𝑙1(𝑡) of the unknown parameters converges to 𝑎=35,𝑏=3,𝑐=12,𝑑=7, and 𝑙=0.5 as 𝑡 tends to 3.

Example 5.3. Let the scaling function be ℎ(𝑡)=1 and the scaling factors are chosen as 𝑚₁=𝑚₂=𝑚₃=𝑚₄=−1. Furthermore, the control gains are chosen as 𝑘₁=𝑘₂=𝑘₃=𝑘₄=3,𝑘₅=𝑘₆=𝑘₇=𝑘₈=𝑘₉=𝑘₁₀=𝑘₁₁=𝑘₁₂=ℎ₁₃=2. Figure 8(a) displays the anti-synchronization between systems (4.1) and (4.2). Figure 8(b) shows the estimates 𝛼₁(𝑡), 𝛽₁(𝑡), 𝛾₁(𝑡), 𝑟₁(𝑡) of the unknown parameters converge to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1as 𝑡→∞. Figure 8(c) shows the estimates 𝑎₁(𝑡), 𝑏₁(𝑡), 𝑐₁(𝑡), 𝑑₁(𝑡), 𝑙₁(𝑡) of the unknown parameters converge to 𝑎=35,𝑏=3,𝑐=12,𝑑=7, and 𝑙=0.5as 𝑡→∞.

(a)

(b)

(c)

Figure 8 

(a) The behaviour of the trajectories 𝑒𝑥, 𝑒𝑦, 𝑒𝑧 and 𝑒𝑤 of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for anti-synchronization. (b): Show the estimates 𝛼1(𝑡), 𝛽1(𝑡), 𝛾1(𝑡), 𝑟1(𝑡) of the unknown parameters converges to 𝛼=10,𝛽=28,𝛾=8/3 and 𝑟=0.1as 𝑡 tends to 3. (c): Show the estimates 𝑎1(𝑡), 𝑏1(𝑡), 𝑐1(𝑡), 𝑑1(𝑡), 𝑙1(𝑡) of the unknown parameters converges to 𝑎=35,𝑏=3,𝑐=12,𝑑=7 and 𝑙=0.5 as 𝑡 tends to 3.

Example 5.4. Let the scaling function be ℎ(𝑡)=sin(0.1𝜋𝑡) and the scaling factors are chosen as 𝑚₁=𝑚₂=𝑚₃=𝑚₄=1. Furthermore, the control gains are chosen as 𝑘₁=𝑘₂=𝑘₃=𝑘₄=3,𝑘₅=𝑘₆=𝑘₇=𝑘₈=𝑘₉=𝑘₁₀=𝑘₁₁=𝑘₁₂=ℎ₁₃=2. Figure 9(a) displays the FPS between systems (4.1) and (4.2). Figure 9(b) shows the estimates 𝛼₁(𝑡), 𝛽₁(𝑡), 𝛾₁(𝑡), 𝑟₁(𝑡) of the unknown parameters converge to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1as 𝑡→∞. Figure 9(c) shows the estimates 𝑎₁(𝑡), 𝑏₁(𝑡), 𝑐₁(𝑡), 𝑑₁(𝑡), 𝑙₁(𝑡) of the unknown parameters converge to 𝑎=35,𝑏=3,𝑐=12,𝑑=7, and 𝑙=0.5as 𝑡→∞.

(a)

(b)

(c)

Figure 9 

(a) The behavior of the trajectories 𝑒𝑥, 𝑒𝑦, 𝑒𝑧,and 𝑒𝑤 of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for FPS. (b) the estimates 𝛼1(𝑡), 𝛽1(𝑡), 𝛾1(𝑡), 𝑟1(𝑡) of the unknown parameters converge to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1 as 𝑡 tends to 3. (c) the estimates 𝑎1(𝑡), 𝑏1(𝑡), 𝑐1(𝑡), 𝑑1(𝑡), 𝑙1(𝑡) of the unknown parameters converge to 𝑎=35,𝑏=3,𝑐=12,𝑑=7, and 𝑙=0.5 as 𝑡 tends to 3.

Example 5.5. Let the scaling function be ℎ(𝑡)=1 and the scaling factors are chosen as 𝑚₁=𝑚₂=𝑚₃=𝑚₄=0.5. Furthermore, the control gains are chosen as 𝑘₁=𝑘₂=𝑘₃=𝑘₄=3,𝑘₅=𝑘₆=𝑘₇=𝑘₈=𝑘₉=𝑘₁₀=𝑘₁₁=𝑘₁₂=ℎ₁₃=2. Figure 10(a) displays the GPS between systems (4.1) and (4.2). Figure 10(b) shows the estimates 𝛼₁(𝑡), 𝛽₁(𝑡), 𝛾₁(𝑡), 𝑟₁(𝑡) of the unknown parameters converge to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1as 𝑡→∞. Figure 10(c) shows the estimates 𝑎₁(𝑡), 𝑏₁(𝑡), 𝑐₁(𝑡), 𝑑₁(𝑡), 𝑙₁(𝑡) of the unknown parameters converges to 𝑎=35,𝑏=3,𝑐=12,𝑑=7, and 𝑙=0.5 as 𝑡→∞.

(a)

(b)

(c)

Figure 10 

(a) The behavior of the trajectories 𝑒𝑥, 𝑒𝑦, 𝑒𝑧, and 𝑒𝑤 of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for GPS. (b) the estimates 𝛼1(𝑡), 𝛽1(𝑡), 𝛾1(𝑡), 𝑟1(𝑡) of the unknown parameters converges to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1 as 𝑡 tends to 3. (c) the estimates 𝑎1(𝑡), 𝑏1(𝑡), 𝑐1(𝑡), 𝑑1(𝑡), 𝑙1(𝑡) of the unknown parameters converges to 𝑎=35,𝑏=3,𝑐=12,𝑑=7, and 𝑙=0.5 as 𝑡 tends to 3.

Example 5.6. Let the scaling function be ℎ(𝑡)=1 and the scaling factors are chosen as 𝑚₁=2,𝑚₂=3,𝑚₃=−2, and 𝑚₄=0.1. Furthermore, the control gains are chosen as 𝑘₁=𝑘₂=𝑘₃=𝑘₄=3,𝑘₅=𝑘₆=𝑘₇=𝑘₈=𝑘₉=𝑘₁₀=𝑘₁₁=𝑘₁₂=ℎ₁₃=2. Figure 11(a) displays the MGPS between systems (4.1) and (4.2). Figure 11(b) show that the estimates 𝛼₁(𝑡), 𝛽₁(𝑡), 𝛾₁(𝑡), 𝑟₁(𝑡) of the unknown parameters converge to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1as 𝑡→∞. Figure 11(c) show that the estimates 𝑎₁(𝑡), 𝑏₁(𝑡), 𝑐₁(𝑡), 𝑑₁(𝑡), 𝑙₁(𝑡) of the unknown parameters converge to 𝑎=35,𝑏=3,𝑐=12,𝑑=7, and 𝑙=0.5 as 𝑡→∞.

(a)

(b)

(c)

Figure 11 

(a) The behaviour of the trajectories 𝑒𝑥, 𝑒𝑦, 𝑒𝑧, and 𝑒𝑤 of the error system between hyperchaotic Lorenz system and hyperchaotic Chen system for MGPS. (b) the estimates 𝛼1(𝑡), 𝛽1(𝑡), 𝛾1(𝑡), 𝑟1(𝑡) of the unknown parameters converge to 𝛼=10,𝛽=28,𝛾=8/3, and 𝑟=0.1 as 𝑡 tends to 3. (c) the estimates 𝑎1(𝑡), 𝑏1(𝑡), 𝑐1(𝑡), 𝑑1(𝑡), 𝑙1(𝑡) of the unknown parameters converge to 𝑎=35, 𝑏=3, 𝑐=12, 𝑑=7, and 𝑙=0.5 as 𝑡 tends to 3.

6. Conclusions

This work investigated modified function projective synchronization between the hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. Based on Lyapunov stability theory, we design adaptive synchronization controllers with corresponding parameter update laws to synchronize the two systems. The MFPS includes complete synchronization, antisynchronization, function projective synchronization (FPS), generalized projective synchronization (GPS), and modified generalized projective synchronization (MGPS). All the theoretical results are verified by numerical simulations to demonstrate the effectiveness of the proposed synchronization schemes. Thus, our synchronization method is successful for some systems with two positive Lyapunov exponents.

Acknowledgment

The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper. The first author acknowledges with thanks the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia for his support this article.