A fractional-order form of a system with stable equilibria and its synchronization

Xiong Wang,Hamid Reza Abdolmohammadi,Adel Ouannas,Viet-Thanh Pham

doi:10.1186/s13662-018-1479-0

Abstract

There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes.

Highlights

There has been a dramatic increase in studying chaos and systems with chaotic behavior in the past decades [1,2,3]
Applications of chaos have been witnessed in various areas ranging from path planning generator [4], secure communications [5,6,7], audio encryption scheme [8], image encryption [9,10,11], to truly random number generator [12, 13]
It has previously been observed that common 3D autonomous chaotic systems, such as Lorenz system [15], Chen system [16], Lü system [17], or Yang’s system [18, 19], have one saddle and two unstable saddle-foci

Summary

Introduction

There has been a dramatic increase in studying chaos and systems with chaotic behavior in the past decades [1,2,3]. It has previously been observed that common 3D autonomous chaotic systems, such as Lorenz system [15], Chen system [16], Lü system [17], or Yang’s system [18, 19], have one saddle and two unstable saddle-foci. Recent evidence suggests that chaos can be observed in 3D autonomous systems with stable equilibria [20, 21]. Several attempts have been made to investigate chaotic systems with stable equilibria. Yang and Chen proposed a chaotic system with one saddle and two stable node-foci [20]. By using the center manifold theory and normal form method, Wei investigated delayed feedback on such a chaotic system with two stable node-foci [22]. It is worth noting that systems with stable equilibria are systems with ‘hidden attractors’ [26,27,28,29]

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Results

Conclusion