A unified fractional action integral framework for memory-dependent circuit dynamics and Josephson junctions

This study presents a unified framework for analyzing electric circuits and Josephson junctions using a fractional action integral that incorporates memory effects and nonlocal behavior. Unlike classical integer-order models, the method extends the action principle to fractional orders, leading to fractional Euler-Lagrange equations that better describe currents, voltages, and phase evolution. It is particularly effective for Josephson junctions, where tunneling currents and phase dynamics show long-term correlations and dissipation. By including fractional-order elements, the framework captures anomalous damping, power-law relaxation, and persistent memory effects in complex circuits. Using a dissipative fractional standard map, the study investigates chaos and the influence of memory on system stability. Numerical results reveal strong sensitivity to fractional parameters, including bifurcations and chaotic attractors. These findings link the classical circuit theory with fractional dynamics, offering new insights for superconducting electronics and the design of nonlinear systems.