On the reducibility of nonlinear differential equations
type: Item
instance of: scholarly article
title: On the reducibility of nonlinear differential equations
zbMATH Open document ID: 0967.34031
published in: Russian Mathematics
publication date: 2001-08-20T00:00:00Z
review text: The author investigates sufficient conditions for the reducibility of equations of the form \[ \frac{dx}{dt} = f(t,x), \] with \( f \in X\), \(f(t,0) \equiv 0, \) and \( X = C^{(p,q)}([T,+\infty) \times\mathbb R^n,\mathbb R^n) \) is the space of all vector functions of dimension \(n,\) which are defined on the set \([T,+\infty) \times\mathbb R^n\), \(p\) times continuously differentiable with respect to variable \(t\), \(p \geq 0,\) and \(q\) times continuously differentiable with respect to the components of the vector \( x\), \(q \geq 1.\) In the first Lyapunov method there is the notion of the Lyapunov transformation and the Erugin theorem (see \textit{B. P. Demidovich} [Lectures of the mathematical theory of stability. Moscow: Nauka (1967; Zbl 0155.41601)] gives the criteria for the reducibility of two linear homogeneous differential equations in the context of the Lyapunov transformation. The author extends the concept of Lyapunov transformation and defines the notion of Lyapunov group of transformations. The definition is used here for the proof of two theorems on the reducibility and one theorem on the stability. The author's theorem 2 contains the Erugin theorem as a special case.
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1542173
zbMATH Keywords: characteristic exponent
zbMATH Keywords: Lyapunov group of transformations
zbMATH Keywords: mutually reducible differential equations
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: Erugin theorem
author: E. V. Voskresenskiĭ
reviewed by: Nikolaj M. Glazunov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1542173