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Bifurcations of nested invariant tori and invariant sets of Lotka-Volterra differential systems

Grant number: 19/05657-4
Support type:Scholarships abroad - Research Internship - Post-doctor
Effective date (Start): May 20, 2019
Effective date (End): March 30, 2020
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Marco Antônio Teixeira
Grantee:Murilo Rodolfo Cândido
Supervisor abroad: Colin Christopher
Home Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Local de pesquisa : Plymouth University, England  
Associated to the scholarship:18/07344-0 - Invariant sets in differential dynamical systems: periodic orbits, invariant tori and algebraic surfaces, BP.PD

Abstract

In the first part of this project, we will study systems in normal form for applying averaging theory. We shall investigate the existence of generic conditions over the averaged system that implies the bifurcation of nested invariant tori in the original system. Our approach to this end will be studying the Chenciner bifurcation in the Poincaré map of the original system.The obtained results will be used for studying the invariant tori bifurcation in some important differential systems like Coullet differential system, The generalized Van der pol-Duffing, and the three-dimensional Lotka-Volterra differential system.In the second part of the project an in-depth study of the three-dimensional Lotka-Volterra differential system will be carried out. Firstly we shall use the method of Darboux to classify the algebraic invariant surfaces of this system and study its integrability. Secondly, we shall investigate the bifurcation of invariant tori. Finally, we are going to use recently obtained results in averaging theory for studying the periodic solutions bifurcating from zero-Hopf equilibria of Lotka-Volterra system.

Scientific publications
(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)
CANDIDO, MURILO R.; LLIBRE, JAUME; VALLS, CLAUDIA. Non-existence, existence, and uniqueness of limit cycles for a generalization of the Van der Pol-Duffing and the Rayleigh-Duffing oscillators. PHYSICA D-NONLINEAR PHENOMENA, v. 407, JUN 2020. Web of Science Citations: 0.
CANDIDO, MURILO R.; NOVAES, DOUGLAS D. On the torus bifurcation in averaging theory. Journal of Differential Equations, v. 268, n. 8, p. 4555-4576, APR 5 2020. Web of Science Citations: 0.

Please report errors in scientific publications list by writing to: cdi@fapesp.br.