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

• Grant number: 19/05657-4
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: May 20, 2019
• End date: March 30, 2020
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Marco Antônio Teixeira
• Grantee: Murilo Rodolfo Cândido
• Supervisor: Colin Christopher
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: 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

(5)

(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.. On the torus bifurcation in averaging theory. Journal of Differential Equations, v. 268, n. 8, p. 4555-4576, APR 5 2020. (18/07344-0, 19/05657-4, 18/16430-8, 18/13481-0, 19/10269-3)

CANDIDO, MURILO R.; NOVAES, DOUGLAS D.; VALLS, CLAUDIA. Periodic solutions and invariant torus in the Rossler system. Nonlinearity, v. 33, n. 9, SEP 2020. (19/05657-4, 19/10269-3, 18/07344-0, 18/16430-8)

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)

CANDIDO, MURILO R.; VALLS, CLAUDIA. Zero-Hopf bifurcation in the general Van der Pol-Duffing equation. JOURNAL OF GEOMETRY AND PHYSICS, v. 179, p. 18-pg., 2022-07-08. (19/05657-4, 18/07344-0)

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. (19/05657-4)