Generating multi-directional hyperchaotic attractors: A novel multi-scroll system based on Julia fractal

Fractals and chaos are two central concepts in the theory of complex systems and play a crucial role in the field of nonlinear dynamics and complexity studies. Despite the relative maturity of research on fractal and chaotic systems, combining them to produce multi-scroll chaotic attractors has received insufficient attention. Firstly, we design a new four-dimensional hyperchaotic system containing the sgn symbolic function, which has no equilibrium point. Next, we delve into the complex dynamical features of this system, via bifurcation diagrams, Lyapunov exponents, and basins of attraction, and we discuss various multistability phenomena. Furthermore, we integrate the Julia fractal process with the new system to create a unique multi-scroll attractor and investigate the effect of the parameters and complex constants in the Julia fractal on the new system. By analyzing the complexity and the Poincaré diagram, we find that the new system incorporating the Julia fractal mapping has richer dynamical properties. Lastly, we performed NIST testing and DSP digital circuit implementation to verify the feasibility and applicability of the new system. The results of this study enrich our understanding of fractals and chaos in practical applications.