Detecting k-hyperbolic orbits and period-doubling bifurcations

• Grant number: 24/04433-3
• Support Opportunities: Scholarships in Brazil - Master
• Effective date (Start): May 01, 2024
• Effective date (End): April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Murilo Rodolfo Cândido
• Grantee: Gean Franco Acuna de La Cruz
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil
• Associated research grant: 23/06076-0 - Averaging based methods for dectecting invariant compact sets, AP.JP

Abstract

In this project, we will study techniques to identify periodic solutions in differential equations using the averaging method. This method has been employed by several authors to establish the number of isolated periodic solutions that may arise in various relevant systems. These bounds are obtained through the analysis of the regular zeros of the mean equations individually. However, we aim to go beyond this traditional approach, seeking to also study the non-isolated zeros of the mean equations that arise in autonomous $n$-dimensional systems ($n\geq3$).To achieve this, we will employ modified versions of the Implicit Function Theorem and Degree Theory to investigate the characteristics of the new periodic orbits, which we call $k$-hyperbolic. Additionally, we will present key specific tools to ensure the existence and stability of these orbits. We emphasize that these orbits are natural candidates to undergo a period-doubling bifurcation. Thus, the presence of these orbits in certain systems may indicate the occurrence of a cascade of period doubling, which can imply the existence of a large number of isolated periodic solutions in the system.