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 researcher: 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
• Research place: 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. (AU)

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.; NOVAES, DOUGLAS D.; VALLS, CLAUDIA. Periodic solutions and invariant torus in the Rossler system. Nonlinearity, v. 33, n. 9 SEP 2020. Web of Science Citations: 0.

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.