Minimal period of solutions to Lipschitz differential equations with\n arbitrary vector norm

2019/11/27 by A. A. Zevin, Zevin, Alexandr
Engineering · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1911.12081

openalex publication_date 2019/11/27 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

The Lipschitz differential equation, x=f(x), in spaces X \∈ Cⁿ and\nX \∈ Rⁿ is considered. The minimal period problem is to find the exact\nlower bound for peri-ods of non-constant solutions, expressed in the Lipschitz\nconstant L. In this paper, some inequality for the components, x_k(t),\nwhich is independent on the space X is found. As a result, it is proved that\nfor any X and n, the normalized minimal period, k=TL \≤ 2\π. In the\nspace Cⁿ, the equality k= 2\π is reached for any X. For Rⁿ, this\nequality is attained for univercally adopted norms.\n

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