Homoclinic Connections and the Melnikov Method in Dynamical Systems.

• Grant number: 26/02339-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: June 01, 2026
• End date: May 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Douglas Duarte Novaes
• Grantee: Ana Paula Schramm Steuernagel
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 24/15612-6 - Ergodic and qualitative theories of dynamical systems III, AP.TEM

Abstract

Homoclinic connections are fundamental structures in Dynamical Systems Theory, as they are associated with the emergence of chaotic behavior. Their detection and characterization constitute central tasks both in the qualitative analysis of differential equations and in the understanding of dynamic mechanisms present in physical, biological, and engineering applications. The Melnikov Method, originally developed for two-dimensional Hamiltonian systems subject to small perturbations, provides a precise analytical criterion to determine whether the separation between the stable and unstable manifolds of a point occurs in the perturbed system, enabling the identification of persistent homoclinic connections and indicating the occurrence of transversal homoclinic points and, therefore, chaotic behavior. This project aims to introduce the student to the geometric and analytical foundations of the theory of homoclinic connections and to present systematically the Melnikov Method in its classical and modern formulation, articulating theory, examples, and current applications. (AU)