Dynamical properties of some classes of interval maps

• Grant number: 22/04040-6
• Support Opportunities: Regular Research Grants
• Start date: July 01, 2022
• End date: June 30, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Márcio Ricardo Alves Gouveia
• Grantee: Márcio Ricardo Alves Gouveia
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• City of the host institution: São José do Rio Preto

Abstract

Poincare's work was the forerunner of modern theory of dynamical systems. Since its introduction, this theory has grown and matured, becoming an important and much studied area of mathematics. Themain objective of this project is to deepen the knowledge in the following areas of dynamical systems:1 - Renormalization Theory and Cherry flows.2 -Rigidity for one-dimensional dynamical systems.In [16] we study some dissipative Lorenz maps of the interval, which are maps of the interval having a discontinuity point and positive derivative (and uniformly) smaller than one at every point of yourdomain. Interested in the dynamics of these maps we study periodic orbits, renormalizations and the invariant minimal singular set when there is no periodic orbit. In a specic set of these maps weprove the existence of a lamination corresponding to the innitely renormalizable maps, as well as the regularity of the leaves of that lamination, in the analytical case. We also have been able to studythe regularity of the conjugations and the holonomy maps of the lamination when the maps have no criticality. In this context we aim to: obtain better properties in the, as for instance, to study thegeneralized Hausdor dimension of the minimal singular set; to extend these results for the case where maps have Lorenz singularity; to study gap maps presenting a plateau in one of its branches; to study gap maps with more than one gap and obtain results similar to [16] for these gap maps with more thanone gap; to study the dynamics of gap maps in the non uniformly contracting case. (AU)

Scientific publications

(The scientific publications listed on this page originate from the Web of Science or SciELO databases. Their authors have cited FAPESP grant or fellowship project numbers awarded to Principal Investigators or Fellowship Recipients, whether or not they are among the authors. This information is collected automatically and retrieved directly from those bibliometric databases.)

PESSOA, CLAUDIO; RIBEIRO, RONISIO; NOVAES, DOUGLAS; GOUVEIA, MARCIO; EUZEBIO, RODRIGO. On cyclicity in discontinuous piecewise linear near-Hamiltonian differential systems with three zones having a saddle in the central one. NONLINEAR DYNAMICS, v. 111, n. 22, p. 23-pg., 2023-11-01. (22/04040-6, 18/13481-0, 19/10269-3, 23/04061-6, 22/09633-5)

GOUVEIA, MARCIO R. A.; LLIBRE, JAUME; ROBERTO, LUCI ANY. Phase Portraits of a Family of Hamiltonian Cubic Systems. DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS, v. N/A, p. 9-pg., 2024-04-05. (22/04040-6, 19/10269-3)

COLLI, EDUARDO; GOUVEIA, MARCIO. HAUSDORFF MEASURES FOR DISSIPATIVE POINCARE MAPS OF CHERRY FLOWS. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v. 47, p. 39-pg., 2025-07-25. (22/04040-6, 19/10269-3)