2001/06/04 by Kayo Ide, K. D. Ide, D. Sornette +1 · 4 citations
Economics, Econometrics and Finance · Environmental Science · Physics and Astronomy · #Complex Systems and Time Series Analysis #Ecosystem dynamics and resilience #Statistical Mechanics and Entropy #cond-mat.stat-mech
paper · pdf · doi:10.1016/s0378-4371(01)00585-4
published as Physica A 307 (1-2), 63-106 (2002) · Latex document of 59 pages including 20 eps figures
arxiv created 2001/06/04 · openalex publication_date 2002/04/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between 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; (3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing.