Piecewise smooth vector fields: Closing Lemmas, shifts and horseshoe dynamics.

• Grant number: 19/10450-0
• Support Opportunities: Regular Research Grants
• Start date: August 01, 2019
• End date: July 31, 2021
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Tiago de Carvalho
• Grantee: Tiago de Carvalho
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil
• City of the host institution: Ribeirão Preto

Abstract

In this research project we will beinterested in studying unpublished topics related to the Qualitative Theory of Ordinary Differential Equations. When we consider smooth vector fields, it is a classic and very relevant research topic, to determine the possible existence of points to which the orbit returns infinite many times in its neighborhood. The 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. In fact, for the smooth case, there are several formulations of Closing Lemmas, where the type of domain or the differentiability of the function used varies. For some of these formulations it is known that the response about 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. With regard to piecewise smooth vector fields, this theme is still little explored and will be the object of study throughout this project. We will look for results related to the possibility (or impossibility) to obtain a Closing Lemma. Furthermore, in the study of such recurrences we will establish conjugations between shifts (with finite and infinite symbols) and flows of piecewise smooth vector fields (via the use of first return maps); besides, we will show dynamics that resemble that obtained in Smale's Horseshoe. (AU)

Scientific publications

(14)

(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.)

DE CARVALHO, TIAGO; CRISTIANO, RONY; GONCALVES, LUIZ FERNANDO; TONON, DURVAL JOSE. Global analysis of the dynamics of a mathematical model to intermittent HIV treatment. NONLINEAR DYNAMICS, v. 101, n. 1, JUL 2020. (17/00883-0, 19/10450-0)

CARVALHO, TIAGO; GONCALVES, LUIZ FERNANDO. Creation of Limit Cycles in Piecewise Smooth Vector Fields Tangent to Nested Tori. Qualitative Theory of Dynamical Systems, v. 20, n. 2, p. 21-pg., 2021-07-01. (19/10269-3, 19/10450-0)

RODRIGUES, DIEGO S.; MANCERA, PAULO F. A.; CARVALHO, TIAGO; GONCALVES, LUIZ FERNANDO. Sliding mode control in a mathematical model to chemoimmunotherapy: The occurrence of typical singularities. Applied Mathematics and Computation, v. 387, DEC 15 2020. (17/00883-0, 19/10450-0)

CARVALHO, TIAGO; CRISTIANO, RONY; RODRIGUES, DIEGO S.; TONON, DURVAL J.. Global Analysis of the Dynamics of a Piecewise Linear Vector Field Model for Prostate Cancer Treatment. JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, OCT 2021. (17/00883-0, 19/10450-0)

CARVALHO, TIAGO; GONCALVES, LUIZ FERNANDO. A FLOW ON S-2 PRESENTING THE BALL AS ITS MINIMAL SET. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, v. 26, n. 8, p. 18-pg., 2021-08-01. (19/10269-3, 19/10450-0)

CARVALHO, TIAGO; CRISTIANO, RONY; RODRIGUES, DIEGO S.; TONON, DURVAL J.. Global Analysis of a piecewise smooth epidemiological model of COVID-19. NONLINEAR DYNAMICS, v. 105, n. 4, AUG 2021. (19/10269-3, 19/10450-0)

FLORENTINO, MARCO; CARVALHO, TIAGO. Piecewise smooth vector fields with sliding motion preserving measure. NONLINEAR ANALYSIS-HYBRID SYSTEMS, v. 57, p. 16-pg., 2025-04-28. (22/02819-6, 19/10269-3, 19/10450-0, 21/12395-6)

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)

CRISTIANO, RONY; PAGANO, DANIEL J.; TONON, DURVAL J.; CARVALHO, TIAGO. Fold bifurcation of T-singularities and invariant manifolds in 3D piecewise-smooth dynamical systems. PHYSICA D-NONLINEAR PHENOMENA, v. 403, FEB 2020. (19/10450-0)

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)

CARVALHO, TIAGO; CUNHA, JACKSON; EUZEBIO, RODRIGO; FLORENTINO, MARCO. Dynamics of an intermittent HIV treatment using piecewise smooth vector fields with two switching manifolds. NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, v. 82, p. 6-pg., 2025-04-01. (21/12395-6, 22/02819-6, 19/10450-0, 19/10269-3)

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)

CARVALHO, TIAGO; NOVAES, DOUGLAS DUARTE; GONCALVES, LUIZ FERNANDO. Sliding Shilnikov connection in Filippov-type predator-prey model. NONLINEAR DYNAMICS, v. 100, n. 3, MAY 2020. (17/00883-0, 19/10450-0, 18/16430-8, 18/13481-0, 19/10269-3)

CARVALHO, TIAGO; EUZEBIO, RODRIGO DONIZETE. Minimal sets and chaos in planar piecewise smooth vector fields. Electronic Journal of Qualitative Theory of Differential Equations, n. 33, p. 1-15, 2020. (17/00883-0, 19/10450-0)