Relaxation Transform for Iterations of the Sinusoidal Map and Its Physical Interpretation as a Memory-Based Process

This research presents a novel mathematical framework for modeling relaxation phenomena in complex systems with memory. We develop a relaxation transformation that continuously interpolates discrete iterations of a sinusoidal mapping, providing a bridge between classical iterative dynamics and continuous relaxation processes. The key innovation lies in interpreting the iteration parameter as a measure of accumulated system "experience" rather than physical time. By introducing a memory function that maps physical time to this experience parameter, the model can describe diverse non-exponential relaxation behaviors commonly observed in glassy materials, biological tissues, and geological media. The approach offers a mathematically elegant alternative to traditional differential equation models, combining computational efficiency with clear physical interpretation. The framework shows particular promise for systems with hierarchical structure, aging phenomena, and multiple relaxation time scales.