Duality structure, asymptotic analysis and emergent fractal sets

Abstract

A new, extended nonlinear framework of the ordinary real analysis  incorporating a novel concept of {\em duality structure} and its applications into  various nonlinear dynamical problems is presented. The duality structure is an asymptotic property that should affect the late time  asymptotic behaviour of a nonlinear dynamical system in a nontrivial way leading naturally to signatures generic to a complex system. We argue
that the present formalism would offer a natural framework to understand the abundance of complex systems in natural, biological, financial and related problems. We show that the power law attenuation of a dispersive, lossy wave equation, conventionally deduced from fractional calculus techniques, could actually arise from the present asymptotic duality structure. Differentiability on a Cantor type fractal set is also formulated.