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The modified function projective synchronization of different dimensional fractional-order chaotic systems with known or unknown parameters is investigated in this paper. Based on the stability theorem of linear fractional-order systems, the adaptive controllers with corresponding parameter update laws for achieving the synchronization are given. The fractional-order chaotic system and hyperchaotic system are applied to achieve synchronization in both reduced order and increased order. The corresponding numerical results coincide with theoretical analysis.

Synchronization has attracted a great deal of interest due to its important applications in ecological systems [

In recent years, the study on the nonlinear dynamics and synchronization control of fractional-order chaotic systems has become a hot topic in nonlinear research area. It is demonstrated that many fractional-order differential systems behave chaotically or hyperchaotically, such as the fractional-order Chua circuit [

At present, many existing schemes focus on synchronization of the fractional-order systems with the same dimension. However, in many real systems, chaos synchronization between different dimensional systems usually occurs, especially in biological and social sciences [

In this paper, we discussed the MFPS problem between different dimensional fractional-order chaotic systems with known or unknown parameters. By reduced order or added order, the problem is transformed to the MFPS between identical dimension chaotic systems. Based on adaptive control method and stability theory of fractional-order systems, an effective controller and a parameter update law for MFPS are proposed by rigorous theoretical analysis. Two groups of examples are considered and their numerical simulations are performed. Numerical simulations verify that MFPS really can occur between different dimension fractional systems.

There are several definitions of fractional derivatives. The Caputo derivative is more popular in the real applications, because the inhomogeneous initial conditions are allowed if such conditions are necessary. The Caputo fractional derivative is defined as

Now, consider the fractional-order drive and response systems are described as system (

When

When the orders of two systems satisfy the condition

In order to unify the order of drive and response system, we can choose arbitrary

When the order of the drive system is lower than that of the response system, we can increase the order of drive system and construct an auxiliary state vector with order

After the above transformation, the dimension of drive system (

When the parameters

For the drive system (

It is easy to see that function projective synchronization and modified projective synchronization are the special cases of modified function projective synchronization where

Next, we will discuss how to choose the controller and the parameter update laws. Let the controller

Then, response system (

According to Definition

Here, we assume that there exists a function

Now, vector

From (

Now the MFPS problem between drive system (

If the elements

Suppose that

Taking conjugate transpose on both sides of the above equation, one can yield

From (

So,

Since

From (

According to the stability theory of fractional-order systems [

The proof is completed.

Therefore, for the drive system (

Theorem

The main difference in the design of the controller between this paper and [

In this section, to demonstrate the effectiveness of the proposed MFPS scheme for different dimension fractional-order chaotic systems, two numerical examples are respectively used to discuss two kinds of cases: the reduced order synchronization with

In this subsection, fractional-order hyperchaotic Lorenz system [

The fractional-order hyperchaotic Lorenz dynamical differential equation can be given by

In order to investigate the reduced-order MFPS behavior between hyperchaotic Lorenz system and Chen system, we assume that the

The controlled fractional-order Chen system with uncertain parameters is defined as

According to the above discussion, we can obtain

Therefore, the possible parameter update law and real matrix

Consequently, it is easy to obtain

Obviously, the elements in

The time evolutions of MFPS errors between system (

Changing parameters

These results show that the MFPS between hyperchaotic system (

It is assumed that the fractional-order chaotic Arneodo system [

The fractional-order Arneodo system is defined as

When

The equations of the 4-dimensional fractional-order Lü system are

It has been shown that system (

Based on the above method, we construct an auxiliary state variable and its fractional derivative

According to the discussion in Section

Now, we can choose

So,

According to the above theorem, the MFPS between 3-dimensional Arneodo system and 4-dimensional Lü system can be achieved via added order. We have performed some numerical simulations to verify the above theoretical analysis. For example, choose

The MFPS errors between added-order system (

In Figure

In summary, the adaptive reduced-order or added-order MFPS of chaotic systems with different dimensions is discussed. When the order of drive system is slightly large, the drive system is order-reduced to synchronize response. If drive system has lower dimension, some auxiliary states are added to increase the dimension of drive system. According to the stability theorems of linear fractional-order systems, an adaptive controller and a parameter update law are presented to synchronize different dimensional systems. It is shown that the response system can not only synchronize with the projection of the drive system but also synchronize with the drive system that includes auxiliary state. This technique has been successfully applied to two examples: fractional-order hyperchaotic Lorenz system drives fractional-order Chen system with unknown parameters; fractional-order Arneodo system drives the fractional-order hyperchaotic Lü system with known parameters.

In this paper, we propose a general method to achieve MFPS of fractional-order chaotic systems even though their dimensions are distinct. The method can be extended to the condition that the derivative orders of drive and response systems are unequal or the fractional-order system with incommensurate orders.

This research was supported by the National Natural Science Foundation of China (Grant nos. 61374178 and 61202085), the Liaoning Provincial Natural Science Foundation of China (Grant no. 201202076), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20120042120010), and the Ph.D. Startup Foundation of Liaoning Province, China (Grant nos. 20111001, 20121001, and 20121002).