Abstract
The fractal and chaos are bound tightly, and their relevant researches are well-established. Few of them, however, concentrate on the research of the possibility of combining fractal and chaotic systems for generating 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, multi-scroll attractors, extreme multi-stability, and initial offset-boosting behavior. The proposed system is combined with fractal transformation to observe a new class of multi-scroll attractors such as multi-ring attractors, separated-scroll attractors, and nested attractors. Particularly, the first swallow-like attractors are found. Moreover, another efficient method to generate a different class of chaotic attractors applies parabola transformation and triangle transformation. Additionally, the spectrum entropy (SE) complexity is also employed to discuss the complexity of the proposed system before and after fractal, resulting in chaotic sequences with the 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.
Highlights
From a computational point of view, the attractors in chaotic systems can be divided into self-excited and hidden attractors
Do we observe swallow-shaped attractors for the first time, but we find another form of attractors based on fractal transformation, the seed system, parabola transformation and triangle transformation
The complex hidden dynamical characteristics of the system are investigated by means of Lyapunov exponents, bifurcation diagrams, phase diagrams, and basin of attraction
Summary
From a computational point of view, the attractors in chaotic systems can be divided into self-excited and hidden attractors. The above-mentioned papers only display multi-scroll attractors without considering both the dynamic analysis and hardware implementation of the fractal system. It is noteworthy that dynamic analysis and hardware implementation were given for the first time in this paper; separated attractors have been neglected. In these foundations, we attempt to combine the non-equilibrium point chaotic system with fractal transformation to generate more complex multi-ring attractors and separated-scroll attractors. 2. The complex dynamical phenomena of the proposed system, such as chaos, hyperchaos, various coexisting multi-scroll chaotic attractors and initial offset boosting, are observed. 3. Combining the hidden attractor with the fractal transformation generates abundant multi-scroll attractors.
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