Integral characterization for Poincaré half-maps in planar linear systems

Abstract

The intrinsic nature of a problem usually suggests a first suitable method to deal with it. Unfortunately, the apparent ease of application of these initial approaches may make their possible flaws seem to be inherent to the problem and often no alternative ways to solve it are searched for. For instance, since linear systems of differential equations are easy to integrate, Poincaré half-maps for piecewise linear systems are always studied by using the direct integration of the system in each zone of linearity. However, this approach is accompanied by two important defects: due to the different spectra of the involved matrices, many cases and strategies must be considered and, since the flight time appears as a new variable, nonlinear complicated equations arise.This manuscript is devoted to present a novel theory to characterize Poincaré half-maps in planar linear systems that avoids the computation of their solutions and the problems it causes. This new perspective rests on the use of line integrals of a specific conservative vector field which is orthogonal to the flow of the linear system. Besides the obvious mathematical interest, this approach is attractive because it allows to simplify the study of piecewise-linear systems and deal with open problems in this field.

Properties of Poincaré half-maps for planar linear systems and some direct applications to periodic orbits of piecewise systems

“…Since the system (2) is linear, the definition of its Poincaré half-maps is usually given in the literature by using the explicit integrated flow (what results in many case-by-case studies) and the intersection points of their orbits with the Poincaré section Σ (what forces the nonlinear implicit appearance of the flight time). Here, we will use a characterization that avoids the explicit computation of the flow (and so the previous flaws), as it is done in [2]. For the sake of completeness, we give a brief summary of the main results and ideas of [2] that are going to be used in this paper.…”

“…Here, we will use a characterization that avoids the explicit computation of the flow (and so the previous flaws), as it is done in [2]. For the sake of completeness, we give a brief summary of the main results and ideas of [2] that are going to be used in this paper.…”

“…This paper relies on a novel characterization of Poincaré half-maps for planar linear systems [2] which allows us to see the properties of these maps from a common point of view and to prove the results in a simple way, without the need of making particularized case-by-case studies. Accordingly, we will not have any of the disadvantages mentioned in the previous paragraphs because this novel characterization does not require integration of the systems and, therefore, the distinction of the spectra of the matrices is not needed.…”

“…The paper is organized as follows. Section 2 presents the integral characterization of Poincaré half-maps for two-dimensional linear systems given in [2]. Two basic consequences of this characterization for the Poincaré half-maps are their analyticity and their understanding as solutions of a differential equation.…”

“…Two basic consequences of this characterization for the Poincaré half-maps are their analyticity and their understanding as solutions of a differential equation. In Section 3, we summarize the results on analyticity of the Poincaré half-maps given in [2] and obtain the Taylor and Newton-Puiseux series expansions at tangency points and infinity by means of the differential equation. Section 4 studies the relationship between the graphs of Poincaré half-maps and the bisector of the fourth quadrant, which is used to establish, in Section 5, the sign of the second derivatives of the Poincaré half-maps.…”