Bifurcation of invariant tori of differential systems via higher order averaging theory

• Grant number: 20/14232-4
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: April 01, 2021
• End date: March 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Douglas Duarte Novaes
• Grantee: Pedro Campos Christo Rodrigues Pereira
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 18/13481-0 - Geometry of control, dynamical and stochastic systems, AP.TEM
• Associated scholarship(s): 23/11002-6 - Averaging theory, bifurcations and catastrophes, BE.EP.DR

Abstract

Invariant manifolds are fundamental tools for understanding the qualitative behaviour of systems. Although the use of the averaging theory to find periodic orbits, a first example of non-trivial invariant sets, is well-established, there is still plenty to be done regarding the study of invariant sets of higher dimensions. In this area, Fenichel's Theory about the persistence of normally hyperbolic invariant manifolds under small perturbation is a well-established tool for the research concerning invariant sets. In this project, both theories, combined with continuation methods, will be employed simultaneously to the study of the existence of invariant tori in regularly perturbed differential systems. The main goal of this project is the generalization of recent results that ensure the existence of invariant sets from the analysis of the averaged system obtained by the averaging method. We hope firstly to establish more generally the correspondence between limit cycles of the averaged system and invariant tori of the original system. More general results arising from the application of the theory of normally hyperbolic invariant manifolds jointly with the averaging theory to the study of differential systems will be investigated subsequently. (AU)

Scientific publications

(4)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

NOVAES, D. D.; PEREIRA, P. C. C. R.. A version of Hilbert's 16th problem for 3D polynomial vector fields: Counting isolated invariant tori. Mathematische Nachrichten, v. 298, n. 2, p. 9-pg., 2024-12-30. (20/14232-4, 19/10269-3, 18/13481-0, 22/09633-5)

PEREIRA, PEDRO C. C. R.; NOVAES, DOUGLAS D.; CANDIDO, MURILO R.. A mechanism for detecting normally hyperbolic invariant tori in differential equations. JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES, v. 177, p. 45-pg., 2023-09-01. (19/05657-4, 22/09633-5, 18/07344-0, 19/10269-3, 21/10606-0, 20/14232-4, 18/13481-0)

NOVAES, DOUGLAS D.; PEREIRA, PEDRO C. C. R.. Invariant tori via higher order averaging method: existence, regularity, convergence, stability, and dynamics. MATHEMATISCHE ANNALEN, v. N/A, p. 48-pg., 2023-06-24. (19/10269-3, 18/13481-0, 20/14232-4, 22/09633-5)

NOVAES, DOUGLAS D.; PEREIRA, PEDRO C. C. R.. On the periodic and antiperiodic aspects of the Floquet normal form. BULLETIN DES SCIENCES MATHEMATIQUES, v. 190, p. 13-pg., 2023-12-19. (19/10269-3, 22/09633-5)