The Markus-Yamabe conjecture does not hold for discontinuous piecewise linear differential systems separated by one straight line
type: Item
instance of: scholarly article
title: The Markus-Yamabe conjecture does not hold for discontinuous piecewise linear differential systems separated by one straight line
zbMATH Open document ID: 1480.34074
author: Jaume Llibre
author: Lucyjane de A. S. Menezes
published in: Journal of Dynamics and Differential Equations
publication date: 2021-05-14T00:00:00Z
review text: The Markus-Yamabe conjecture [\textit{L. Markus} and \textit{H. Yamabe}, Osaka Math. J. 12, 305--317 (1960; Zbl 0096.28802)] states that if a differentiable system \(\dot{x}=f(x)\) has a singularity \(x^*\in\mathbb{R}^n\), \(f(x^*)=0\), and all eigenvalues of the Jacobian matrix \(Df(x)\) have negative real part for any \(x\in\mathbb{R}^n\), then \(x^*\) is a global attractor. The conjecture is true for dimension \(n=1,2\) and for linear systems. There are counterexamples for polynomial vector fields in \(\mathbb{R}^n\) with \(n>2\). The authors consider discontinuous piecewise linear differential systems of the form \[ \dot{z}= \begin{cases} X(z),& \text{if } x>0, \\ Y(z),& \text{if } x<0, \end{cases} \] where \(z=(x,y)\in\mathbb{R}^2\), \(X\) and \(Y\) are linear vector fields, the real part of the eigenvalues of \(DX(z)\) and \(DY(z)\) are negative, \(Y(0)=0\), and the singularity of \(X\) leaves in the half-plane \(\{x<0\}\). The authors call this kind of piecewise linear systems as \textit{piecewise linear Markus-Yamabe system} and prove that the Markus-Yamabe conjecture does not hold for them.
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 7348188
zbMATH Keywords: discontinuous piecewise linear differential system
zbMATH Keywords: Markus-Yamabe conjecture
reviewed by: Oleg Anashkin
MaRDI profile type: Publication
full work available at URL: s10884-020-09825-8
OpenAlex ID: W3003688214
Wikidata QID: Q122961876
cites work: Q3129010
cites work: Weakened Markus--Yamabe conditions for 2-dimensional global asymptotic stability
cites work: A polynomial counterexample to the Markus-Yamabe conjecture
cites work: A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization
cites work: Q3994508
cites work: Global stability of equilibria
cites work: Chebyshev-Haar systems in the theory of discontinuous Kellogg kernels
cites work: A polynomial class of Markus–Yamabe counterexamples and examples in ℝ3
cites work: A solution to the bidimensional global asymptotic stability conjecture
cites work: Maximum number of limit cycles for certain piecewise linear dynamical systems
cites work: Q3275105
cites work: Q3820959
cites work: On exponential decay and the Markus-Yamabe conjecture in infinite dimensions with applications to the Cima system
DOI: 10.1007/S10884-020-09825-8
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description: scientific article; zbMATH DE number 7348188