Averaging principle and a variational approach to the problem of bifurcation of periodic solutions from nonisolated equilibria of the averaged equation
type: Item
instance of: scholarly article
title: Averaging principle and a variational approach to the problem of bifurcation of periodic solutions from nonisolated equilibria of the averaged equation
zbMATH Open document ID: 1264.34089
DOI: 10.1134/S0012266112030032
author: B. A. Mikhailenko
author: Mikhail Kamenskii
published in: Differential Equations
publication date: 2012-08-13T00:00:00Z
review text: Consider the differential equation \[ {dx\over dt}= Ax+ f\Biggl({t\over\varepsilon}, x\Biggr),\tag{\(*\)} \] where \(A\) is a nonsingular \(n\times n\)-matrix, \(f\in C(\mathbb{R}\times\mathbb{R}^n,\mathbb{R}^n)\) is \(T\)-periodic in the first variable and twice differentiable in the second variable, \(0\leq \varepsilon\ll 1\). Under the additional assumptions (A1), the averaged equation \[ {dx\over dt}= Ax+{1\over T} \int^T_0 f(t,x)\,dx \] has a one-dimensional connected manifold of equilibria \(\{x_0(\theta): \theta\in [0,1]\}\), where \(x_0\) is twice differentiable, and it holds \(x_0'(\theta)\neq 0\,\forall\theta\), and (A2), the zero eigenvalue of the operator \[ A+{1\over T} \int^T_0 {\partial f\over\partial x} (t,x_0(\theta))\,dt \] is simple, and by using the abstract Malkin bifurcation function \(M\) defined in the paper, the authors prove the following averaging principle. Theorem. Let the assumptions (A1) and (A2) be satisfied. Additionally it is assumed that there is \(\theta_0\in [0,1]\) such that \(M(\theta_0)= 0\), \(M'(\theta_0)= 0\). Then, for sufficiently small \(\varepsilon\), there exists a family of \(\varepsilon T\)-periodic solutions \(\{x(\varepsilon)\}\) of \((*)\) such that \(x(\varepsilon)\to x_0(\theta_0)\) as \(\varepsilon\to 0\). Moreover, a variational approach is presented to solve the same problem.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 6065258
Wikidata QID: Q115252180
MaRDI profile type: Publication
cites work: Periodic solutions of a perturbed autonomous system
cites work: Q5541565
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description: scientific article; zbMATH DE number 6065258