Dynamics of piecewise linear Li\'enard systems near infinity
- 1Fuzhou Univ., Fuzhou, Fujian, China
- 2Central South Univ., Changsha, Hunan, China
Abstract
In this article, we study the dynamics near infinity for a class of piecewise linear Lienard systems dx/dt=y−F(x), dy/dt=−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.
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BibTeX
@article{jia2026dynamics,
title = {{Dynamics of piecewise linear Li\'enard systems near infinity}},
author = {Man Jia and Yuhao Meng},
journal = {Electronic Journal of Differential Equations},
year = {2026},
month = sep,
url = {http://ejde.math.txstate.edu/Volumes/2026/74/abstr.html},
note = {CC BY},
}
RIS
TY - JOUR
TI - Dynamics of piecewise linear Li\'enard systems near infinity
AU - Man Jia
AU - Yuhao Meng
PY - 2026
DA - 2026/09/01
JO - Electronic Journal of Differential Equations
UR - http://ejde.math.txstate.edu/Volumes/2026/74/abstr.html
AB - In this article, we study the dynamics near infinity for a class of piecewise linear Lienard systems dx/dt=y−F(x), dy/dt=−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.
SN - 1072-6691
ER -