Crossover behavior for a periodic chaotic model in its fractal and mirror symmetrical design

Abstract

Dynamical systems related to damped and driven oscillators and those with both periodic and chaotic oscillations have proven to be fascinating for many researchers around the world who are dealing with non-linear oscillatory models. Adding to theses processes, a mirror symmetrical structure can only be more captivating since their applications in sciences and engineering are widely increasing. That is why in this paper, we combine two dynamical structures, namely the mirror symmetrical structure and the fractal effect. The mirror symmetrical aspect is analyzed by inserting into the system an additional term, a quadratic function involving a duality-symmetric multi-segment. Secondly, the fractal effect is analyzed by inserting into the system the fractal–fractional operator with the power law kernel. The final model reveals the existence of the so-called fixed-point chaotic attractor involved in a self-duplication process via the mirror-fractal process. The final model is also able to generate a new type of mirror fractal dynamical process with y-oriented crossover structure and both chaotic periodic and chaotic oscillations. As we gradually change some of the model’s parameters , the lower and higher parts of the attractors are shown to be moving away from the crossover junction line due to the fractal effect. The results show another way of considering some dynamical models related to damped and driven oscillators.