Abstract
The topic related to the coexistence of different synchronization types between fractional-order chaotic systems is almost unexplored in the literature. Referring to commensurate and incommensurate fractional systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between nonidentical systems characterized by different dimensions and different orders. In particular, the paper shows that identical synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS) coexist when synchronizing a three-dimensional master system with a fourth-dimensional slave system. The approach, which can be applied to a wide class of chaotic/hyperchaotic fractional-order systems in the master-slave configuration, is based on two new theorems involving the fractional Lyapunov method and stability theory of linear fractional systems. Two examples are provided to highlight the capability of the conceived method. In particular, referring to commensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Rössler system of order 2.7 and the hyperchaotic four-dimensional Chen system of order 3.84. Finally, referring to incommensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Lü system of order 2.955 and the hyperchaotic four-dimensional Lorenz system of order 3.86.
Highlights
By starting from the milestone by Pecora and Carroll [1], over the last years, great efforts have been devoted to the study of chaos synchronization in dynamical systems described by integer-order differential equations and difference equations [2]
This increased complexity related to both the number of synchronization types and the capability to synchronize chaotic dynamics with hyperchaotic ones and provides a deeper insight into the synchronization phenomena between systems described by fractional differential equations
7 Conclusions In this paper, we have presented a new approach to rigorously study the coexistence of some synchronization types between fractional-order chaotic systems characterized by different dimensions and different orders
Summary
By starting from the milestone by Pecora and Carroll [1], over the last years, great efforts have been devoted to the study of chaos synchronization in dynamical systems described by integer-order differential equations and difference equations [2]. By considering the errors in (23), it can be concluded identical synchronization (IS), anti-phase synchronization (AS) and inverse full state hybrid projective synchronization (IFSHPS) coexist when synchronizing the chaotic fractional-order commensurate Rössler system (19) and the hyperchaotic fractional-order commensurate Chen system (21). Theorem 4 Given the error system (29) between the master system (7) and the slave system (8), identical synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS) coexist if the following control law is taken:. By considering the errors in (40) it can be concluded that identical synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS) coexist when synchronizing the chaotic fractional-order incommensurate Lü system (35) and the hyperchaotic fractional-order incommensurate Lorenz system (37)
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