Dynamic Analysis of Complex Synchronization Schemes between Integer Order and Fractional Order Chaotic Systems with Different Dimensions

1Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, 12002 Tebessa, Algeria 2Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, China 3Hanoi University of Science and Technology, Hanoi, Vietnam 4Department of Matter Sciences, University of Larbi Tebessi, Tebessa, Algeria 5Active Devices and Materials Laboratory (LCAM), Larbi Ben M’hidi University, Oum El Bouaghi, Algeria


Introduction
Nature is intrinsically nonlinear.So, it is not surprising that most of the systems we encounter in the real world are nonlinear.And what is interesting is that some of these nonlinear systems can be described by fractional order differential equations which can display a variety of behaviors including chaos and hyperchaos [1][2][3][4][5].Recently, study on synchronization of fractional order chaotic systems has started to attract increasing attention of many researchers [6][7][8][9][10][11][12], since the synchronization of chaotic systems with integer order is understood well and widely explored [13][14][15].Many scientists who are interested in the field of chaos synchronization have struggled to achieve the synchronization between integer order and fractional order chaotic systems.
At present, many schemes of control have been proposed to study the problem of synchronization between integer order and fractional order chaotic systems such as anticipating synchronization [16], function projective synchronization [17], complete synchronization [18], antisynchronization [19], Q-S synchronization [20], and generalized synchronization [21].Also, different techniques have been introduced to synchronize integer order and fractional order chaotic systems.For example, a nonlinear feedback control method has been introduced in [22].The idea of tracking control has been applied in [23,24].In [25], general control scheme has been described.A new fuzzy sliding mode method has been proposed in [26], and a sliding mode method has been designed in [27,28].Synchronization of a class of hyperchaotic systems has been studied in [29].A practical method, based on circuit simulation, has been presented in [30], and in [31] a robust observer technique has been tackled.
Complete synchronization (CS), projective synchronization (PS), full state hybrid function projective synchronization (FSHFPS), and generalized synchronization (GS) are effective approaches to achieve synchronization and have been used widely in integer order chaotic systems [32][33][34][35] and fractional order chaotic systems [36][37][38][39].Studying inverse problems of synchronization is an attractive and important idea.Recently, some interesting types of synchronization have 2 Complexity been introduced such as inverse projective synchronization (IPS) [40], inverse full state hybrid projective synchronization (IFSHPS) [41], inverse full state hybrid function projective synchronization (IFSHFPS) [42], and inverse generalized synchronization (IGS) [43].Coexistence of several types of synchronization produces new complex type of chaos synchronization.Not long ago, many approaches for the problem of coexistence of synchronization types have been proposed in discrete time chaotic systems, integer order chaotic systems, and fractional order chaotic systems [44][45][46][47].The coexistence of different type of synchronization is very useful in secure communication and chaotic encryption schemes.
This paper introduces new approaches to study the coexistence of some types of synchronization between integer order and fractional order chaotic systems with different dimensions.The new results, derived in this paper, are established in the form of simple conditions about the linear parts of the slave system and the master system, respectively, which are very convenient to verify.Using fractional Lyapunov approach, the coexistence of complete synchronization (CS), projective synchronization (PS), full state hybrid function projective synchronization (FSHFPS), and generalized synchronization (GS) between integer order master system and fractional order slave system is investigated.Based on integer order Lyapunov method, the coexistence of inverse projective synchronization (IPS), inverse full state hybrid function projective synchronization (IFSHFPS), and inverse generalized synchronization (IGS) between fractional order master system and integer order slave system is studied.The capability of the approaches is illustrated by numerical examples.
The rest of this paper is arranged as follows.Some theoretical bases of fractional calculus are introduced in Section 2. In Section 3, our main results of the paper are presented.In Section 4, our approaches are applied between some typical chaotic and hyperchaotic systems to show the effectiveness of the derived results.Section 5 is the conclusion of the paper.

Fractional Integration and Derivative.
There are several definitions of fractional derivatives [48,49].The two most commonly used are the Riemann-Liouville and Caputo definitions.Each definition uses Riemann-Liouville fractional integration and derivatives of whole order.The difference between the two definitions is in the order of evaluation.The Caputo derivative is a time domain computation method [50].In real applications, the Caputo derivative is more popular since the unhomogenous initial conditions are permitted if such conditions are necessary [51,52].Furthermore, these initial values are prone to measure since they all have idiographic meanings.The Riemann-Liouville fractional integral operator of order  ≥ 0 of the function () is defined as Some properties of the operator   can be found, for example, in [53].In this study, Caputo definition is used and the fractional derivative of () is defined as for  − 1 <  ≤ ,  ∈ N,  > 0. The fractional differential operator    is left-inverse (and not right-inverse) to the fractional integral operator   ; that is,      = , where  is the identity operator.

Fractional Order Lyapunov Stability.
Consider the following fractional order system: where () = ( 1 (),  2 (), . . .,   ())  ,  is a rational number between 0 and 1 and    is the Caputo fractional derivative of order , and  : R  → R  .For stability analysis of fractional order systems, a fractional extension of Lyapunov direct method has been proposed by the following theorem.
Theorem 2 (see [55]).If there exists a positive definite Lyapunov function (()) such that    (()) < 0, for all  > 0, then the trivial solution of system ( 4) is asymptotically stable.Now, we present a new lemma which is helpful in the proving analysis of the proposed method.
Error system (7), between master system (5) and slave system (6), can be differentiated as follows: Error system (8) can be described as where Rewrite error system (9) in the compact form where Theorem 5.There exists a suitable feedback gain matrix  ∈ R 4×4 to realize the coexistence of CS, PS, FSHFPS, and GS between master system (5) and slave system (6) under the following control law: Proof.Substituting ( 12) into (11), one can have If a Lyapunov function candidate is chosen as (()) = (1/2)  ()(), then, the time Caputo fractional derivative of  along the trajectory of system ( 13) is as follows: and using Lemma 3 in ( 14) we get If we select the feedback gain matrix  such that  −  is a negative definite matrix, then we get    (()) < 0. From Theorem 2, the zero solution of system ( 13) is a globally asymptotically stable; that is, lim →+∞   () = 0,  = 1, 2, 3, 4. We conclude that master system (5) and slave system (6) are globally synchronized.
To achieve synchronization between systems ( 16) and ( 17), we assume that  is an invertible matrix and  −1 its inverse matrix.Hence, we have the following result.Theorem 7. IPS, IFSHFPS, and IGS coexist between master system (16) and slave system (17) under the following conditions: Proof.By substituting the control law (ii) into ( 22), the error system can be written as Using (iii), then we get V(()) < 0. From Lemma 1, the zero solution of error system ( 24) is globally asymptotically stable and therefore, master system (16) the slave system (17) are globally synchronized.

Numerical Examples
In this section, two numerical examples are considered to validate the proposed chaos synchronization schemes.
The projections of the hyperchaotic Lorenz attractor are shown in Figure 2.

Conclusion
This paper has presented new schemes to study the coexistence of some types of chaos synchronization between nonidentical and different dimensional master and slave systems described by integer order and fractional order differential equations.The first scheme was constructed by combining CS, PS, FSHFPS, and GS in the synchronization of 3D integer order master system and 4D fractional order slave system.The second one was proposed when IPS, IFSHFPS, and IGS coexist between 3D fractional order master system and 4D integer order slave system.By exploiting fractional order Lyapunov approach and integer order Lyapunov method, the proposed synchronization approaches were rigorously proved to be achievable.The capability of the methods was illustrated by numerical examples and computer simulations.