Necessary and sufficient conditions for the boundedness of solutions to two-dimensional quadratic systems in a positively invariant half-plane
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for the boundedness of solutions to two-dimensional quadratic systems in a positively invariant half-plane
zbMATH Open document ID: 1203.34060
DOI: 10.1134/S1064562410010102
published in: Doklady Mathematics
publication date: 2010-11-17T00:00:00Z
review text: The author studies the planar system \[ \frac{dx}{dt}= x^2+xy+y, \quad \frac{dy}{dt}= a_2x^2+ b_2xy+c_2y^2+ \alpha_2x+\beta_2y \tag{1} \] under the conditions \[ c_2\neq 0, \qquad c_2\neq-1, \qquad c_2\neq b_2-a_2. \] He shows that the half-plane \(\Gamma:= \{x>-1\), \(y\in\mathbb R\}\) is positively invariant under the flow of (1) and that (1) can be reduced to a Liénard system. By this way the author is able to derive conditions on the coefficients of (1) such that {\parindent=7mm \begin{itemize}\item[(i)] All trajectories of (1) staring in \(\Gamma\) are bounded; \item[(ii)] System (1) has a limit cycle in \(\Gamma\); \item[(iii)] System (1) has two limit cycle sin \(\Gamma\). \end{itemize}}
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5816854
author: G. A. Leonov
MaRDI profile type: Publication
cites work: Limit cycles of the Liénard equation with discontinuous coefficients
cites work: HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
cites work: Computation of focus values with applications
full work available at URL: s1064562410010102
OpenAlex ID: W2028155051
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description: scientific article; zbMATH DE number 5816854