| Recent years,research on the frontier issues such as chaos and fractional calculus has shifted from pure mathematical theory to engineering application.Fractional-order multi-scroll chaotic system has the characteristics of simple structure,complicated dynamic behavior,multi-scroll,high sensitivity,and it has the advantage of high space,high dynamic storage capacity,anti-decipher and anti-interference when used as information encryption key.Therefore,fractional-order multi-scroll chaotic system may have great application prospects in the field of information security.This paper focuses on the design and implementation of several fractional-order(FO)multi-scroll chaotic systems,and has obtained the following results:(1)A FO one-directional multi-scroll chaotic system based on stair function series is designed.Firstly,a new FO three-dimensional linear system is constructed,controller based on stair function series is added to system to extend the equilibrium point with index 2,and one-directional multi-scroll attractor is observed in the FO chaotic system.The dynamic behavior is analyzed through stability theory of equilibrium,Lyapunov exponents and Poincaré map,confirming the chaotic behavior of the designed system.Moreover,The physical circuit of the constructed system is given and the experimental results are consistent with the simulation results,which verifies the validity and feasibility of the method.(2)A FO one-directional multi-scroll hyper-chaotic system based on hysteresis function series is designed.Hysteresis function series are added to a new designed four-dimensional FO linear system,equilibrium points of index 2 in the system are extended and multi-scroll attractors are successfully obtained in the FO system.With the help of theoretical tools such as Lyapunov exponents and Poincaré map,the hyper-chaotic behavior in the system is confirmed.Furthermore,a circuit is designed for the hyper-chaotic system and the circuit simulation results agree with numerical simulation,which shows the effectiveness and feasibility of the method.(3)A FO two-directional multi-scroll chaotic system based on combined functions is designed.Based on a new three-dimensional FO linear system,a controller composed of saturated function sequence and stair function sequence is added to the FO system to extend equilibrium points in different axes,respectively,and grid multi-scroll attractors emerge in the system.The stability of the equilibrium point is analyzed,Lyapunov exponent and the Poincaré section numerical methods are used to reveal the chaos generation mechanism of the multi-scroll chaotic system.Circuit simulation results are presented,which are consistent with the numerical simulation results,proving the correctness and feasibility of our design.(4)A systemic method is proposed to construct FO hyper-chaotic system family,and a class of FO hyper-chaotic system are obtained based on the method.Firstly,a new three-dimensional FO linear system is constructed and dimension of the system is increased by introducing extra variables according to the rules designed,equilibrium points with index 2 are extended by introducing stair function series into the system,thus,the system can generate hyper-chaotic multi-scroll of any high dimension.The dynamic behavior is analyzed through stability theory of equilibrium,Lyapunov exponents and Poincaré map,confirming the hyper-chaotic behavior of the designed system.A circuit is designed for a four dimensional FO system in this FO system family,the circuit simulation results are in good agreement with the numerical simulation results,which further illustrate the correctness and validity of the designed rules. |
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