The fractal and chaos are bound tightly, and their relevant researches are well-established. Few of them, however, concentrates on the research of the possibility of combining the fractal and the chaotic systems to generate multi-scroll chaotic attractors. This paper presents a novel non-equilibrium point chaotic system, exhibiting extremely rich and complex hidden behaviors including chaos, hyper-chaos, multiscroll attractors, extreme multi-stability and initial offset-boosting. The proposed system is combined with fractal transformation respectively, and a new class of multi-scroll attractors, such as multi-ring attractors and separated-scroll attractors, is observed. Particularly, swallow-shaped attractors for the first time is found. Moreover, another efficient method to generate a different class of chaotic attractors uses parabola transformation and triangle transformation. Additionally, the spectrum entropy (SE) complexity is employed to discuss the complexity of the proposed system before and after fractal, resulting in a chaotic sequences with fractal transformation that has higher complexity. Finally, we develop a hardware platform to implement the presented attractors before and after fractal in a way to confirm the accuracy of the numerical simulations, providing a theoretical basis for the next application in image encryption.
Over the past few decades, the research of dissipative chaotic systems has yielded many achievements in both theory and application. However, attractors in dissipative systems are easily reconstructed by the attacker, which leads to information security problems. Compared with dissipative systems, conservative ones can effectively avoid these reconstructing attacks due to the absence of attractors. Therefore, conservative systems have advantages in chaos-based applications. Currently, there are still relatively few studies on conservative systems. For this purpose, based on the simplest memristor circuit in this paper, a non-Hamiltonian 3D conservative system without equilibria is proposed. The phase volume conservatism is analyzed by calculating the divergence of the system. Furthermore, a Kolmogorov-type transformation suggests that the Hamiltonian energy is not conservative. The most prominent property in the conservative system is that it exhibits quasi-periodic 3D tori with heterogeneous coexisting and different amplitude rescaling trajectories triggered by initial values. In addition, the results of Spectral Entropy analysis and NIST test show that the system can produce pseudo-random numbers with high randomness. To the best of our knowledge, there is no 3D conservative system with such complex dynamics, especially in a memristive conservative system. Finally, the analog circuit of the system is designed and implemented to test its feasibility as well.
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