2014/09/09 by Anatoly Manita, Manita, Anatoly
Computer Science · Decision Sciences · #60G51 #60G55 #60J25 #60K05 #60K35 #FOS: Electrical engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Network Time Synchronization Technologies #Nonlinear Dynamics and Pattern Formation #Probability (math.PR) #Simulation Techniques and Applications #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1409.2919
openalex publication_date 2014/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose stochastic N-component synchronization models (x1(t),...,xN(t)), xj∈ℝd, t∈ℝ+, whose dynamics is described by Levy processes and synchronizing jumps. We prove that symmetric models reach synchronization in a stochastic sense: differences between components dkj(N)(t)=xk(t)-xj(t) have limits in distribution as t→∞. We give conditions of existence of natural (intrinsic) space scales for large synchronized systems, i.e., we are looking for such sequences \bN\ that distribution of dkj(N)(∞)/bN converges to some limit as N→∞. It appears that such sequence exists if the Levy process enters a domain of attraction of some stable law. For Markovian synchronization models based on α-stable Levy processes this results holds for any finite N in the precise form with bN=(N-1)1/α. For non-Markovian models similar results hold only in the asymptotic sense. The class of limiting laws includes the Linnik distributions. We also discuss generalizations of these theorems to the case of non-uniform matrix-based intrinsic scales. The central point of our proofs is a representation of characteristic functions of dkj(N)(t) via probability distribution of a superposition of N independent renewal processes.