Recently, the notion of hidden extreme multistability and hidden attractors is very attractive in chaos theory and nonlinear dynamics. In this paper, by utilizing a simple state feedback control technique, a novel 4D fractional-order hyperchaotic system is introduced. Of particular interest is that this new system has no equilibrium, which indicates that its attractors are all hidden and thus Shil’nikov method cannot be applied to prove the existence of chaos for lacking hetero-clinic or homo-clinic orbits. Compared with other fractional-order chaotic or hyperchaotic systems, this new system possesses three unique and remarkable features: (i) The amazing and interesting phenomenon of the coexistence of infinitely many hidden attractors with respect to same system parameters and different initial conditions is observed, meaning that hidden extreme multistability arises. (ii) By varying the initial conditions and selecting appropriate system parameters, the striking phenomenon of antimonotonicity is first discovered, especially in such a fractional-order hyperchaotic system without equilibrium. (iii) An attractive special feature of the convenience of offset boosting control of the system is also revealed. The complex and rich hidden dynamic behaviors of this system are investigated by using conventional nonlinear analysis tools, including equilibrium stability, phase portraits, bifurcation diagram, Lyapunov exponents, spectral entropy complexity, and so on. Furthermore, a hardware electronic circuit is designed and implemented. The hardware experimental results and the numerical simulations of the same system on the Matlab platform are well consistent with each other, which demonstrates the feasibility of this new fractional-order hyperchaotic system.

A novel three-dimensional fractional-order autonomous chaotic system marked by the ample and complex coexisting attractors is presented. There are a total of seven terms including four nonlinearities in the new system. The evolution of coexisting attractors of the system are numerically investigated by considering both the fractional-order and other system parameters as bifurcation parameters. Numerical simulation results indicate that the system has a huge amount of multifarious coexisting strange attractors for various ranges of parameters, including coexisting point, periodic attractors, multifarious coexisting chaotic, and periodic attractors. Compared with other chaotic systems, the biggest difference and most attractive feature is the capability of the proposed fractional-order system to produce coexisting attractors that undergo a simultaneous displacement phenomenon with variation of a single parameter. Moreover, it is worth noting that constant Lyapunov exponents and the interesting phenomenon of transient coexisting attractors are also observed. Finally, the corresponding implementation circuit is designed. The consistency of the hardware experimental results with numerical simulations verifies the feasibility of the new fractional-order chaotic system.

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