On limit cycles in piecewise linear vector fields with algebraic discontinuity variety

• Grant number: 21/10606-0
• Support Opportunities: Scholarships abroad - Research
• Start date: May 01, 2022
• End date: July 31, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Douglas Duarte Novaes
• Grantee: Douglas Duarte Novaes
• Host Investigator: Joan Torregrosa I. Arus
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Universitat Autònoma de Barcelona (UAB), Spain
• Associated research grant: 18/13481-0 - Geometry of control, dynamical and stochastic systems, AP.TEM

Abstract

The second part of the Hilbert's sixteenth problem consists in determining the upper bound H(n) for the number of limit cycles that planar polynomial vector fields of degree n can have. For n greater than or equal 2, it is still unknown whether H(n) is finite or not. The main achievements obtained so far establish lower bounds for H(n). Regarding asymptotic behavior, the best result says that H(n) grows as fast as n^2 log(n). Better lower bounds for small values of n are known in the research literature. In the recent paper "Some open problems in low dimensional dynamical systems" by A. Gasull, Problem 18 proposes another Hilbert's sixteenth type problem, namely improving the lower bounds for L(n), which is defined as the maximum number of limit cycles that planar piecewise linear vector fields with two zones separated by a branch of an algebraic curve of degree n can have. So far, the best known general lower bound for L(n) is [n/2]. Again, better lower bounds for small values of n are known in the research literature. Providing upper bounds for L(n) has also been shown to be very challenging, even in the linear case, that is L(1). So far, it is still open whether L(1) is finite or not. Accordingly, the main goals of this project are: 1. To improve the lower bounds for L(n). Here, it is conjectured that L(n) grows as fast as n^2. The main techniques that is going to be used to approach this problem are: a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold; Chebyshev theory for ECT-systems; and Pseudo-Hopf Bifurcation. 2. To obtain an upper bound for L(1). The main technique that is going to be used to approach this problem is an integral characterization of the half-return map for linear systems. (AU)

Scientific publications

(8)

(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)

NOVAES, DOUGLAS D.; SILVA, LEANDRO A.. ON THE NON-EXISTENCE OF ISOCHRONOUS TANGENTIAL CENTERS IN FILIPPOV VECTOR FIELDS. Proceedings of the American Mathematical Society, v. 150, n. 12, p. 10-pg., 2022-07-01. (18/13481-0, 19/10269-3, 21/10606-0)

CARMONA, VICTORIANO; FERNANDEZ-SANCHEZ, FERNANDO; NOVAES, DOUGLAS D.. Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region. COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, v. 123, p. 18-pg., 2023-04-28. (19/10269-3, 21/10606-0, 18/13481-0, 22/09633-5)

CARMONA, VICTORIANO; FERNANDEZ-SANCHEZ, FERNANDO; NOVAES, DOUGLAS D.. A Succinct Characterization of Period Annuli in Planar Piecewise Linear Differential Systems with a Straight Line of Nonsmoothness. JOURNAL OF NONLINEAR SCIENCE, v. 33, n. 5, p. 13-pg., 2023-10-01. (19/10269-3, 21/10606-0, 18/13481-0, 22/09633-5)

NOVAES, DOUGLAS D.. AN AVERAGING RESULT FOR PERIODIC SOLUTIONS OF CARATHEODORY DIFFERENTIAL EQUATIONS. Proceedings of the American Mathematical Society, v. 150, n. 7, p. 10-pg., 2022-07-01. (21/10606-0, 19/10269-3, 18/16430-8)

CARMONA, VICTORIANO; FERNANDEZ-SANCHEZ, FERNANDO; GARCIA-MEDINA, ELISABETH; NOVAES, DOUGLAS D.. Properties of Poincare half-maps for planar linear systems and some direct applications to periodic orbits of piecewise systems. Electronic Journal of Qualitative Theory of Differential Equations, v. N/A, n. 22, p. 18-pg., 2023-01-01. (19/10269-3, 21/10606-0, 18/13481-0, 22/09633-5)

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)

NOVAES, DOUGLAS D.. On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set. PHYSICA D-NONLINEAR PHENOMENA, v. 441, p. 15-pg., 2022-12-01. (19/10269-3, 18/13481-0, 21/10606-0)

CARMONA, VICTORIANO; FERNANDEZ-SANCHEZ, FERNANDO; NOVAES, DOUGLAS D.. Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. Applied Mathematics Letters, v. 137, p. 8-pg., 2023-03-01. (19/10269-3, 21/10606-0, 18/13481-0, 22/09633-5)