From Biological Fractal Dynamics to Analog Fractional-Order Circuits: A System Design Framework Based on Operatorization Thought

Fractional calculus, with its unique advantage in describing memory and nonlocal effects, has become a key tool for modeling nonlinear dynamic systems. Its integration with circuit systems has opened up a new paradigm for modern circuit design. However, the development of analog fractional-order circuits (AFCs) is limited by the difficulty in physical implementation of ideal fractional-order elements (fractances) and the lack of systematic design theories. Based on biological dynamics in physical fractal space and operatorization thought, this article aims to provide a novel approach to address the above challenges. The core work of this paper is to systematically introduce operatorization thought into the field of circuits and construct a universal fractional-order system design framework applicable to multiple disciplines. Unlike traditional methods relying on backend optimization, this research focuses on the intrinsic construction of frontend fractional-order systems. By deeply understanding the physical connotation of fractional-order operators and leveraging the force-electricity analogy principle, the approximation of fractional-order circuits using classical integer-order components is realized. Theoretical analysis and circuit simulations demonstrate that this method not only simplifies the design process but also reveals the inherent connection between fractional-order circuits and fractal operators from a physical perspective, providing a solid theoretical basis and innovative path for the design of high-performance, integrable fractal circuits.