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

Author(s)

Carmona, Victoriano ; Fernandez-Sanchez, Fernando ; Garcia-Medina, Elisabeth ; Novaes, Douglas D.

• Document type: Journal article
• Source: Electronic Journal of Qualitative Theory of Differential Equations; v. N/A, n. 22, p. 18-pg., 2023-01-01.

Abstract

This paper deals with fundamental properties of Poincare half-maps defined on a straight line for planar linear systems. Concretely, we focus on the analyticity of the Poincare half-maps, their series expansions (Taylor and Newton-Puiseux) at the tangency point and at infinity, the relative position between the graph of Poincare 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 piece-wise linear systems. Accordingly, we also provide some of their most immediate, but non-trivial, consequences regarding periodic orbits. (AU)

• FAPESP's process: 19/10269-3 - Ergodic and qualitative theories of dynamical systems II
• Grantee: Claudio Aguinaldo Buzzi
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 21/10606-0 - On limit cycles in piecewise linear vector fields with algebraic discontinuity variety
• Grantee: Douglas Duarte Novaes
• Support Opportunities: Scholarships abroad - Research
• FAPESP's process: 18/13481-0 - Geometry of control, dynamical and stochastic systems
• Grantee: Marco Antônio Teixeira
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 22/09633-5 - Averaging theory for studying invariant tori and periodic behavior in differential equations and inclusions
• Grantee: Douglas Duarte Novaes
• Support Opportunities: Regular Research Grants