Multi-directional annular multi-wing chaotic system based on Julia fractals

Abstract

Fractals and chaos are inextricably linked, and relevant study has been done on both. However, the studies that combine fractals and chaotic systems to produce multi- scroll or multi-wing chaotic attractors are all extend multi-scroll or multi-wing in the even direction. To this end, by improving the traditional fractal mapping, a chaotic system that can extend multiple wings in any direction of odd and even is designed and analyzed. The main point of this method is as follows: Firstly, a new family of even-symmetric segmented composite functions is proposed and applied to a family of generalized Lorenz systems, which can generate multi-wing chaotic systems in two directions and the number of wings in each direction can also be increased with the parameter N. Secondly, the improved Julia fractal is applied to two-direction multi-wing chaotic systems to generate a class of single-direction multi-wing chaotic systems. Finally, the combination of this class of single-direction multi-wing chaotic systems and higher-order Julia fractals is used to realize chaotic systems with extended multi-wing in odd and even arbitrary directions. Computer simulations confirm the method's viability. In addition, the maximum Lyapunov exponent, bifurcation diagram, Poincare diagram, and PE complexity chaos diagram are also analyzed as examples of multi-directional multi-wing Lü chaotic systems, demonstrating the system's superiority in odd directions over even directions and the existence and effective scalability of multi-directional multi-wing chaotic systems.