Theory of self-similar oscillatory finite-time singularities in Finance, Population and Rupture

TL;DR

This paper introduces a simple two-dimensional dynamical system that models finite-time singularities with oscillations, applicable to finance, population dynamics, and material failure, revealing self-similar fractal structures in phase space.

Contribution

It presents a fundamental equation capturing oscillatory finite-time singularities across diverse systems, highlighting the role of nonlinear feedback and phase space spiral structures.

Findings

Models stock market price dynamics with nonlinear feedback.

Describes population growth with finite carrying capacity and nonlinear effects.

Explains material failure with competing damage and healing processes.

Abstract

This is a short letter summarizing the long paper cond-mat/0106047 in which we present a simple two-dimensional dynamical system reaching a singularity in finite time decorated by accelerating oscillations due to the interplay between nonlinear positive feedback and reversal in the inertia. This provides a fundamental equation for the dynamics of (1) stock market prices in the presence of nonlinear trend-followers and nonlinear value investors, (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity and (3) the failure of a material subject to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing. The rich fractal scaling properties of the dynamics are traced back to the self-similar spiral structure in phase space unfolding around an unstable spiral point at the origin.