Self-Replication in a Mathematical Model of Particle Movement in Fluids, with Its Hardware Implementation

The use of fractals and chaos theory has opened many new research possibilities in science, engineering, and technology. This paper adds to these studies by using numerical methods, analytical techniques, and experimental circuit components to study and simulate a modified stretch–twist–fold (STF) flow. This flow is important for understanding how particles move in fluids and is related to the dynamo theorem. We first check the basic properties of the system to ensure it is well defined. Then, we create a numerical method, analyze its errors, and run simulations that include the fractal–fractional operators. These simulations show how the system combines self-replication with chaotic behavior for different parameter values. We also build the system on a Field-Programmable Gate Array (FPGA) board, which produces results similar to the numerical simulations. This research helps improve our understanding of fractals and chaos theory and their role in explaining the complex fluid dynamics, especially the STF flow and its link to particle movement in fluids connected to the dynamo theorem.