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Constructing Countably Many Distinct Suslin Sets of Characteristic Exponents in the Perron Effect of Change of Their Values
Differential Equations ( IF 0.6 ) Pub Date : 2021-01-21 , DOI: 10.1134/s00122661200120022
N. A. Izobov , A. V. Il’in

Abstract

For any parameters \(m>1\), \(\lambda _1\le \lambda _2<0 \), and \(\varepsilon >0 \) and for two sequences \(\{S_{in}\} \) of uniformly bounded arbitrary Suslin sets \(S_{1n}\subset [\lambda _1+\varepsilon ,b_1]\) and \(S_{2n}\subset [\max \{\lambda _2+\varepsilon ,b_1\},b_2]\), we prove the existence of a two-dimensional nonlinear differential system with a linear approximation that has characteristic exponents \(\lambda _1\) and \(\lambda _2 \) and with a disturbance of the \(m \)th order of smallness in a neighborhood of the origin and possible growth outside it such that all nontrivial solutions of this system are infinitely extendible and have finite Lyapunov exponents. For any \(n\in \mathbb {N} \), these exponents form the following sets: \(S_{1n} \) for solutions with initial values \((c_1,0)\ne 0 \), where \(|c_1|\in (n-1,n] \), and \(S_{2n} \) for solutions with initial values \((c_1,c_2) \) where \(|c_2|\in (n-1,n] \). In particular, for any bounded Suslin sets \(S_{-}\subset (-\infty ,0)\) and \(S_{+}\subset (0,+\infty ) \) we have also established the existence of a nonlinear system whose Lyapunov exponents for all nontrivial solutions form these two sets (singletons in the Perron case).



中文翻译:

在值变化的Perron效应中构造数量众多的不同Suslin特征指数集

摘要

对于任何参数\(m> 1 \)\(\ lambda _1 \ le \ lambda _2 <0 \)\(\ varepsilon> 0 \)以及两个序列\(\ {S_ {in} \} \ )的均等边界任意Suslin集 \(S_ {1n} \ subset [\ lambda _1 + \ varepsilon,b_1] \)\(S_ {2n} \ subset [\ max \ {\ lambda _2 + \ varepsilon,b_1 \}, b_2] \),我们证明了存在一个线性近似的二维非线性微分系统,该系统具有特征指数\(\ lambda _1 \)\(\ lambda _2 \),并且具有\(m \ )起源附近的小阶数及其外部可能的增长,使得该系统的所有非平凡解都是无限可扩展的,并且具有有限的Lyapunov指数。对于任何\(n \ in \ mathbb {N} \),这些指数形成以下集合: \(S_ {1n} \)用于初始值为\((c_1,0)\ ne 0 \)的解,其中\ (| c_1 | \ in(n-1,n] \)\(S_ {2n} \)对于初始值为\((c_1,c_2)\)的解,其中\(| c_2 | \ in(n- 1,n] \)特别是对于任何有界Suslin集 \(S _ {-} \ subset(-\ infty,0)\)\(S _ {+} \ subset(0,+ \ infty)\) 我们还建立了非线性系统的存在,该系统的所有非平凡解的李雅普诺夫指数均由这两组(佩隆案例中的子集)组成。

更新日期:2021-01-21
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