In this research work, a dual of Julia transformation method following a nested approach is proposed to generate m × k 2 wings in chaotic systems and enhance their dynamics from chaos to fractals. The designed complex systems are further coupled with the differential drive system representing a two-wheeled mini robot. The provided pseudocode is tested on various chaotic systems to check its compatibility, but for analytical results, the Halvorsen chaotic system is considered as an example. Moreover, the dynamics of the original (seed) system and its prototype forms are discussed with the aid of phase portraits, time series, bifurcation diagrams, and their corresponding Lyapunov exponents. A fractal is the advanced form of a chaotic system and embeds complexity into system trajectories. Therefore, the transformed systems are merged with a robotic equation to observe the impact of fractals in other fields. Additionally, a new type of dynamic obstacle that follows an irregular path is designed in the current study. Finally, the analytical findings on fractals and their robotic applications are compared with existing literature to underscore the significance of the proposed work. All fractal simulations and robotic experiments a are conducted using MATLAB.
Khan et al. (Tue,) studied this question.