Abstract
•Control technique combined to Julia’s & fractal-fractional processes are used.•Auto-replication in systems of chaotic attractors is generated.•Such chaotic systems of attractors contain merged basins of attraction.•Symmetry property and the number of basins of attraction are conserved.•Merged basins of attraction can move due to the fractional dynamics’ impact.
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.