Closing lemmas and shifts for piecewise smooth vector fields

• Grant number: 17/18255-6
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)
• Start date: January 01, 2018
• End date: February 28, 2021
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Tiago de Carvalho
• Grantee: Andre do Amaral Antunes
• 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
• Associated research grant: 13/24541-0 - Ergodic and qualitative theory of dynamical systems, AP.TEM

Abstract

In a given dynamic system it is possible to have points to which the orbit returns infinitely many times in its neighborhood. Closing lemma seeks to establish when perturbations of the initial system have a periodic orbit and thus the orbit "closes", hence the name of the lemma. Throughout this project we will study several formulations of Closing lemma, where the type of domain or the differentiability of the functions used are varied. For some of these formulations it is known that the response to the existence of the closed orbit is positive, for other formulations it is known that the answer is negative and there are still other formulations where there is no definitive answer. The student will focus on obtaining results related to versions of Closing lemma (where it is possible and where it is impossible to establish?) for piecewise smooth vector fields. Moreover, in the study of such recurrences we will establish conjugations between shifts (of finite and infinite symbols) and first return maps, in this way, we will obtain ergodic results regarding piecewise smooth vector fields. (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.)

ANTUNES, A. A.; CARVALHO, T.; GOMIDE, O. M. L.. Closing Lemma for piecewise smooth vector fields with a recurrent point. NONLINEAR ANALYSIS-HYBRID SYSTEMS, v. 53, p. 9-pg., 2024-04-09. (19/10450-0, 17/18255-6, 21/12395-6, 19/10269-3, 22/02819-6)

CARVALHO, TIAGO; ANTUNES, ANDRE DO AMARAL. Symbolic dynamics of planar piecewise smooth vector fields. Journal of Differential Equations, v. 419, p. 25-pg., 2024-11-29. (19/10269-3, 19/10450-0, 21/12395-6, 17/18255-6, 22/02819-6, 17/00883-0)

ANTUNES, ANDRE AMARAL; CARVALHO, TIAGO; VARAO, REGIS. On topological entropy of piecewise smooth vector fields. Journal of Differential Equations, v. 362, p. 22-pg., 2023-03-13. (19/10450-0, 16/22475-9, 19/10269-3, 21/12395-6, 17/18255-6, 17/06463-3)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

ANTUNES, Andre do Amaral. Piecewise smooth vectors fields: closing lemma, topological entropy and Shifts. 2021. Doctoral Thesis - Universidade Estadual Paulista (Unesp). Instituto de Biociências Letras e Ciências Exatas. São José do Rio Preto São José do Rio Preto.