[Paper Review] Duality Structure, Asymptotic analysis and Emergent Fractal sets

Dhurjati Prasad Datta, Soma Sarkar|arXiv (Cornell University)|Jan 12, 2016

Complex Systems and Time Series Analysis3 citations

TL;DR

This paper introduces a duality structure in asymptotic analysis that extends real analysis to model complex, non-differentiable systems via a non-Archimedean extension of the real line. It shows that power-law wave attenuation and differentiability on Cantor-type fractals emerge naturally from this duality, providing a novel framework for modeling complex systems in physics, biology, and finance.

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.

Motivation & Objective

• To develop a new nonlinear framework of real analysis incorporating duality structure as an asymptotic property affecting late-time behavior of nonlinear systems.
• To address the limitations of standard real analysis in describing complex systems such as turbulent flows, chaotic attractors, and financial time series.
• To formulate differentiability on fractal sets like the Cantor set using a duality-invariant, non-Archimedean extension of the real line.
• To show that power-law wave attenuation—typically derived via fractional calculus—can instead emerge from the proposed duality structure.
• To establish a duality-invariant jump differential calculus applicable to continuous but non-differentiable functions on fractal sets.

Proposed method

• Introduces an asymptotic visibility metric with duality invariance that remains non-zero in arbitrarily small neighborhoods of a point, even as the standard Lebesgue measure vanishes.
• Extends the real line ℝ to a structured, soft real number system ℝ* via a non-Archimedean extension that incorporates the duality transformation x ↦ X̃⁻¹ = |x|⁻ˢ for 0 < |x| < 1 and s > 0.
• Applies a fundamental selection principle to ensure continuity of measure functions across gaps in fractal constructions, such as the Cantor set.
• Uses self-similarity and rescaling to derive a scale-invariant differential equation dfc(X)/dX = 1 on the prolongation of 0, leading to the Cantor staircase function.
• Defines a jump derivation DJ equivalent to an ordinary derivative under a deformed measure dX(x), enabling Riemann-Stieltjes integration on fractal spaces.
• Models first-order differential equations on fractals via jump differential equations of the form DJy = χC(x)f(x,y), where χC is the characteristic function of the Cantor set.

Experimental results

Research questions

• RQ1 How can a duality structure in asymptotic analysis lead to nontrivial signatures of complex systems in nonlinear dynamical systems?
• RQ2 Can power-law wave attenuation in dispersive, lossy media be derived without fractional calculus, using only asymptotic duality structure?
• RQ3 How can differentiability be defined on nowhere dense, measure-zero fractal sets like the Cantor set within a generalized calculus framework?
• RQ4 What role does the duality-invariant visibility metric play in enabling non-Archimedean extensions of the real line?
• RQ5 How does the jump differential calculus on fractals relate to standard calculus via a deformed measure dX(x)?

Key findings

• The duality structure leads to a non-Archimedean extension ℝ* of the real line ℝ, where a duality-invariant visibility metric preserves non-zero measure in arbitrarily small neighborhoods of a point.
• Power-law wave attenuation, conventionally derived via fractional calculus, is shown to emerge naturally from the asymptotic duality structure without requiring fractional derivatives.
• Differentiability on the Cantor set is formulated via a jump derivation DJ, which acts as an ordinary derivative under a deformed measure dX(x), enabling a generalized calculus on fractals.
• The Cantor staircase function fc(X) satisfies the scale-invariant equation dfc(X)/dX = 1 on the prolongation of 0, with Hausdorff dimension s = log 2 / log 3.
• The jump differential equation DJy = χC(x)f(x,y) models systems that vary only on fractal sets, with the dynamics equivalent to an ordinary differential equation in the renormalized variable X.
• Riemann-Stieltjes integration becomes available on fractal spaces through the deformed measure dX(x), generalizing standard integration theory to irregular sets.