Properties of Poincar\'{e} half-maps for planar linear systems and some direct applications to periodic orbits of piecewise systems

TL;DR

This paper investigates the fundamental properties of Poincaré half-maps for planar linear systems, focusing on their analyticity, series expansions, and geometric relations, to better understand periodic orbits in piecewise systems.

Contribution

It provides new insights into the analyticity, series expansions, and geometric properties of Poincaré half-maps, crucial for analyzing periodic orbits in piecewise linear systems.

Findings

Poincaré half-maps are analytic except at certain points.

Series expansions at tangency points and infinity are characterized.

Properties of the half-maps influence the existence of periodic orbits.

Abstract

This paper deals with fundamental properties of Poincar\'e half-maps defined on a straight line for planar linear systems. Concretely, we focus on the analyticity of the Poincar\'e half-maps, their series expansions (Taylor and Newton-Puiseux) at the tangency point and at infinity, the relative position between the graph of Poincar\'e half-maps and the bisector of the fourth quadrant, and the sign of their second derivatives. All these properties are essential to understand the dynamic behavior of planar piecewise linear systems. Accordingly, we also provide some of their most immediate, but non-trivial, consequences regarding periodic orbits.