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


In this research project we will beinterested in studying unpublished topics related to the Qualitative Theory of Ordinary Differential Equations.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)

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Scientific publications (7)
(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)
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, . (19/10450-0)
CARVALHO, TIAGO; NOVAES, DOUGLAS DUARTE; GONCALVES, LUIZ FERNANDO. Sliding Shilnikov connection in Filippov-type predator-prey model. NONLINEAR DYNAMICS, v. 100, n. 3, . (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, . (19/10450-0, 17/00883-0)
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, . (19/10450-0, 17/00883-0)
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, . (17/00883-0, 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, . (19/10269-3, 19/10450-0)
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, . (17/00883-0, 19/10450-0)

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