2020/10/31 by Spiros Cotsakis
Computer Science · Mathematics · Physics and Astronomy · #Chaos control and synchronization #Combinatorics #Computer science #Function (biology) #Geometry #Mathematical analysis #Mathematics #Nonlinear Dynamics and Pattern Formation #Physics #Point (geometry) #Quantum chaos and dynamical systems #Singularity #Statistical physics #Synchronization (alternating current) #Topology (electrical circuits) #Unison #gr-qc #hep-th
paper · pdf · doi:10.1098/rsta.2021.0189
published as Phil. Trans. R. Soc. A380 (2022) 20210189 · 40 pages, version to appear in the Phil. Trans. Roy. Soc. A
arxiv created 2021/07/19 · openalex publication_date 2022/03/14 · arxiv updated 2022/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the problem of asymptotic synchronization of different spatial points coupled to each other in inhomogeneous spacetime and undergoing chaotic Mixmaster oscillations towards the singularity. We demonstrate that for couplings larger than some threshold value, two Mixmaster spatial points A,B, with A in the past of B, synchronize and thereby proceed in perfect unison towards the initial singularity. We further show that there is a Lyapunov function for the synchronization dynamics that makes different spatial points able to synchronize exponentially fast in the past direction. We provide an elementary proof of how an arbitrary spatial point responds to the mean field created by the oscillators, leading to their direct interaction through spontaneous synchronization. These results ascribe a clear physical meaning of early-time synchronization leading to a resetting effect for the two BKL maps corresponding to two distinct oscillating spatial points, as the two maps converge to each other to become indistinguishable at the end of synchronization. Our results imply that the universe generically organizes itself through simpler, synchronized, states as it approaches the initial singularity. A discussion of further implications of early-time inhomogeneous Mixmaster synchronization is also provided.