Man Jia, Yuhao Meng
Abstract:
In this article, we study the dynamics near infinity for a class of piecewise
linear Lienard systems
\({\mathrm{d}x}/{\mathrm{d}t}=y-F(x)\), \({\mathrm{d}y}/{\mathrm{d}t}=-g(x)\),
with \(n\) parallel switching lines, where \(F(x)\) and \(g(x)\) may be continuous or discontinuous. Using
the qualitative theory of differential equations, we obtain a complete classification of the phase portraits near infinity in the
Poincare disc. More precisely, when equilibria exist on the equator of the
Poincare disc, we show that the dynamics near infinity are determined by the two
outermost linear systems. When no equilibrium exists on the equator,
a center-focus problem arises at infinity. In this case, we provide sufficient
conditions for the dynamics near infinity to be a stable focus or an unstable focus,
and derive a necessary condition for the dynamics near infinity to be a center.
Furthermore, for the continuous case where the system has a unique finite equilibrium,
we establish a global center criterion.
Submitted March 6, 2026. Published September 28, 2026.
Math Subject Classifications: 34C07, 34D20, 37C20.
Key Words: Piecewise linear Lienard system; Poincare compactification; center-focus problem.
DOI: 10.58997/ejde.2026.74
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| Man Jia School of Mathematics and Statistics Fuzhou University Fuzhou 350116, Fujian, China email: manjia@fzu.edu.cn |
| Yuhao Meng School of Mathematics and Statistics HNP-LAMA, Central South University Changsha 410083, Hunan, China email: yuhao_meng@csu.edu.cn |
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