On the diversity of Poincaré mappings for cubic equations with variable coefficients
type: Item
instance of: scholarly article
title: On the diversity of Poincaré mappings for cubic equations with variable coefficients
zbMATH Open document ID: 0955.34010
DOI: 10.1007/BF02467118
author: A. A. Panov
publication date: 2001-02-12T00:00:00Z
review text: Consider the scalar differential equation \[ dx/dt = a_3 (t)x^3 + a_2 (t) x^2 + a_1 (t)x + a_0 (t) \tag \(*\) \] for \( t \in [0,1]\) with \(a_i \in C^1\) for \(i=1,2,3\). Let \(\varphi (t, x)\) be the solution to \((*)\) satisfing \(\varphi (0,x)=x\), by \(\pi (x): = \varphi (1,x)\) we denote the Poincaré map of \((*)\). A solution \(\varphi\) to \((*)\) satisfying \(\varphi (1,x)=x\) is called a cyclic solution. By studying the set of Poincaré maps, the author proves that \((*)\) can have arbitrarily many (nondegenerate) cyclic solutions.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1452683
zbMATH Keywords: Poincaré map
zbMATH Keywords: cyclic solutions
MaRDI profile type: Publication
cites work: On the number of solutions of the equation \(\sum^n_{j=0}a_j(t)x^j,0\leq t\leq 1\), for which \(x(0)=x(1)\)
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1452683