2001/11/09 by S. Gluzman, Simon Gluzman, Didier Sornette · 1 citation
Computer Science · Economics, Econometrics and Finance · Physics and Astronomy · #Complex Systems and Time Series Analysis #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.66.016134
published as Physical Review E 6601 N1 PT2:U315-U328 (2002) · Latex document of 18 pages + 7 ps figures
arxiv created 2001/11/09 · openalex publication_date 2002/07/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Starting from a representation of the early time evolution of a dynamical system in terms of the polynomial expression of some observable phi(t) as a function of the time variable in some interval 0 < or = t < or = T, we investigate how to extrapolate/forecast in some optimal stability sense the future evolution of phi(t) for time t>T. Using the functional renormalization of Yukalov and Gluzman, we offer a general classification of the possible regimes that can be defined based on the sole knowledge of the coefficients of a second-order polynomial representation of the dynamics. In particular, we investigate the conditions for the occurrence of finite-time singularities from the structure of the time series, and quantify the critical time and the functional nature of the singularity when present. We also describe the regimes when a smooth extremum replaces the singularity and determine its position and amplitude. This extends previous works by (1) quantifying the stability of the functional renormalization method more accurately, (2) introducing more global constraints in terms of moments, and (3) going beyond the "mean-field" approximation.