Novel Excitation of local fractional dynamics

TL;DR

This paper introduces a new analytical framework based on asymptotic duality to systematically transition from classical integral order dynamics to local fractional dynamics on fractal spaces, with applications to wave equations.

Contribution

It presents a novel concept of asymptotic duality structure enabling continuous deformation of classical dynamics into fractional dynamics on fractal spaces.

Findings

Duality structure relates to renormalization group transformations.

Invariant derivation operator under duality enables fractional derivatives.

Application to wave equations shows meaningful fractal dynamics results.

Abstract

The question of a possible excitation and emergence of fractional type dynamics, as a more realistic framework for understanding emergence of complex systems, directly from a conventional integral order dynamics, in the form a continuous transition or deformation, is of significant interest. Although there have been a lot of activities in nonlinear, fractional or not, dynamical systems, the above question appears yet to be addressed systematically in the current literature. The present work may be considered to be a step forward in this direction. Based on a novel concept of asymptotic duality structure, we present here an extended analytical framework that would provide a scenario for realizing the above stated continuous deformation of integral order dynamics to a local fractional order dynamics on a fractal and fractional space. The related concepts of self dual and strictly dual asymptotics are introduced and there relevance in connection with smooth and nonsmooth deformation of the real line are pointed out. The relationship of the duality structure and renormalization group is examined. The ordinary derivation operator is shown to be invariant under this duality enabled renormalization group transformation, leading thereby to a {\em natural} realization of local fractional type derivative in a fractal space. As an application we discuss linear wave equation in one and two dimensions and show how the underlying integral order wave equation could be deformed and renormalized suitably to yield meaningful results for vibration of a fractal string or wave propagation in a region with fractal boundary.