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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

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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

Rights: Copyright © 2014 The Belgian Mathematical Society

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Vol.21 • No. 4 • october 2014
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