Existence of an Anti-Perron Effect of Change of Positive Exponents of the Linear Approximation System to Negative Ones under Perturbations of a Higher Order of Smallness

Pub Date : 2024-02-26 DOI:10.1134/s0012266123120029

N. A. Izobov, A. V. Il’in

引用次数: 0

Abstract

We prove the existence of a two-dimensional linear system \(\dot {x}=A(t)x \), \(t\geq t_0\), with bounded infinitely differentiable coefficients and all positive characteristic exponents, as well as an infinitely differentiable \(m\)-perturbation \(f(t,y) \) having an order \(m>1 \) of smallness in a neighborhood of the origin \(y=0 \) and an order of growth not exceeding \(m \) outside it, such that the perturbed system \(\dot {y}=A( t)y+\thinspace f(t,y)\), \(y\in \mathbb {R}^2 \), \(t\geq t_0\), has a solution \(y(t) \) with a negative Lyapunov exponent.

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线性近似系统的正指数在高阶小扰动下变为负指数的反珀伦效应的存在性

Abstract We prove existence of a two-dimensional linear system \(\dot {x}=A(t)x \), \(t\geq t_0\), withbounded infinitely differentiable coefficients and all positive characteristic exponents, as well as aninfinitely differentiable \(m\)-perturbation\(f(t,y) \) having an order \(m>. 1\) in the neighborhood of origin\(y=0\) with an smallness and an order growth not exceed\(m\) outside it；在原点（y=0）的邻域内有一个小的增长阶次，而在它之外有一个不超过（m）的增长阶次、such that the perturbed system\(\dot {y}=A( t)y+\thinspace f(t,y)\),\(y\in \mathbb {R}^2 \), \(t\geq t_0\), has asolution \(y(t) \) with a negative Lyapunov exponent.

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