Largely recognized as leading concepts in network traffic prediction, machining or image processing, the processes of auto-duplication, self-organization and auto-replication are highly useful and fascinating for chaos and fractal theorists. Those processes appear naturally around us as observed on trees, river deltas, lightning, growth spirals, flowers, romanesco broccoli, frost, etc. Using mathematical notions, concepts and processes to reproduce and control auto-duplication and auto-replication dynamics have attracted the attention of scientists and engineers given their wide range of applications. We use the control technique combined to two different concepts, Julia’s process and fractal-fractional operator, to generate auto-replication in systems of chaotic attractors with two and three merged basins of attraction. The systems used here comprise a controller part, namely the switching-manifold control. Both systems are solved numerically using Julia’s scheme and Legendre wavelets methods. Numerical simulations reveal a fascinating capability for the systems to auto-replicate while conserving the initial properties of the systems, such as symmetry and number of basins of attraction. The merged basins of attraction are shown to be moving away as the result of the fractional dynamics’ impact. Great results that may interest scientists and engineers dealing with fractals and chaos.

自动复制，自组织和自动复制的过程在网络流量预测，机加工或图像处理中被公认为领先概念，对混乱和分形理论家来说非常有用且令人着迷。在树木，河流三角洲，闪电，生长螺旋，花朵，罗马花椰菜，霜冻等观察到的过程中，这些过程自然地出现在我们周围。使用数学概念，概念和过程来复制和控制自动复制和自动复制的动态吸引了人们。科学家和工程师的广泛应用引起了人们的关注。我们将控制技术与茱莉亚过程和分形分数算子这两个不同的概念结合使用，以在具有两个和三个合并的吸引盆的混沌吸引子系统中生成自动复制。在此使用的系统包括控制器部分，即开关歧管控制。两种系统均使用Julia方案和Legendre小波方法进行数值求解。数值模拟显示了系统自动复制的迷人能力，同时保留了系统的初始属性，例如对称性和吸引盆数量。由于分数动力学的影响，合并后的吸引力盆地正在逐渐消失。处理分形和混沌的科学家和工程师可能会感兴趣的出色结果。数值模拟显示了系统自动复制的迷人能力，同时保留了系统的初始属性，例如对称性和吸引盆数量。由于分数动力学的影响，合并后的吸引力盆地正在逐渐消失。处理分形和混沌的科学家和工程师可能会感兴趣的出色结果。数值模拟显示了系统自动复制的迷人能力，同时保留了系统的初始属性，例如对称性和吸引盆数量。由于分数动力学的影响，合并后的吸引力盆地正在逐渐消失。处理分形和混沌的科学家和工程师可能会感兴趣的出色结果。