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october 2014 Piecewise Linear Systems with Closed Sliding Poly-Trajectories
Jaime R. de Moraes, Paulo R. da Silva
Bull. Belg. Math. Soc. Simon Stevin 21(4): 653-684 (october 2014). DOI: 10.36045/bbms/1414091008

## Abstract

In this paper we study piecewise linear (PWL) vector fields $$F(x,y)=\left\{\begin{array}{l} F^+(x,y)\quad if \quad x\geq0, \\ F^-(x,y)\quad if \quad x\leq0,\end{array}\right.$$ where $\mathrm{x}=(x,y)\in \mathbb{R}^2$, $F^{+}(\mathrm{x})=A^{+}\mathrm{x}+b^{+}$ and $F^{-}(\mathrm{x})=A^{-}\mathrm{x}+b^{-}$, $A^{+}=(a_{ij}^{+})$ and $A^{-}=(a_{ij}^{-})$ are $(2\times2)$ constant matrices, $b^{+}=(b_{1}^{+},b_{2}^{+})\in\mathbb{R}^2$ and $b^{-}=(b_{1}^{-},b_{2}^{-})\in\mathbb{R}^2$ are constant vectors in $\mathbb{R}^2$. We suppose that the equilibrium points are saddle or focus in each half-plane. We establish a correspondence between the PWL vector fields and vectors formed by some of the following parameters: sets on $\Sigma$ (crossing, sliding or escaping), kind of equilibrium (real or virtual), intersection of manifolds with $\Sigma$, stability and orientation of the focus. Such vectors are called configurations. We reduce the number of configurations by an equivalent relation. Besides, we analyze for which configurations the corresponding PWL vector fields can have or not closed sliding poly-trajectories.

## Citation

Jaime R. de Moraes. Paulo R. da Silva. "Piecewise Linear Systems with Closed Sliding Poly-Trajectories." Bull. Belg. Math. Soc. Simon Stevin 21 (4) 653 - 684, october 2014. https://doi.org/10.36045/bbms/1414091008

## Information

Published: october 2014
First available in Project Euclid: 23 October 2014

zbMATH: 1341.37009
MathSciNet: MR3271326
Digital Object Identifier: 10.36045/bbms/1414091008

Subjects:
Primary: 34H99 , 37C10

Keywords: Piecewise linear systems , poly-trajectories , vector fields