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Oncological therapies usually are applied intermittently, i.e., not continuously over time. In the periods in between, though, the cancer cells are usually left free to grow. This intermittency is a key issue in prostate cancer hormonal therapies based on androgen suppression. Here, we address this treatment modality by analyzing a piecewise smooth vector field approach. In fact, using the PSA (prostate-specific antigen) serum level as a control variable to switch between treatment and no-treatment periods of hormone therapy (androgen withdrawal), by means of typical parameter values, our theoretical analysis supports the idea that intermittent androgen suppression may prevent a prostate cancer relapse for a specific class of patients.

In the 1990s Teixeira studied discontinuous piecewise-smooth (DPWS) dynamical systems in R 3 , whose state space is divided in two open regions by a plane acting as the switching manifold, and presenting two lines of quadratic tangency, one for each involved vector field. We call T-singularity (Teixeira singularity) the point of transverse intersection of the quadratic tangency lines that are of the invisible type. In this paper, we analyze DPWS systems that present two branches of T-singularities. In our approach we consider the cases where either a pseudo-equilibrium collides to just one of the two T-singularities or the two T-singularities (and also a pseudo-equilibrium between them) collides in a single point. As a consequence of these bifurcations, until now not covered by the literature, we can observe the birth of limit cycles, determine their stability and provide an upper limit for the number of limit cycles that occur in this scenario.

The main purpose of this paper is to analyze the Hopf-like bifurcations in 3D piecewise linear systems. Such bifurcations are characterized by the birth of a piecewise smooth limit cycle that bifurcates from a singular point located at the discontinuity manifold. In particular, this paper concerns systems of the form $${\dot{x}}=Ax+b^{\pm }$$ which are ubiquitous in control theory. For this class of systems, we show the occurrence of two distinct types of Hopf-like bifurcations, each of which gives rise to a crossing limit cycle (CLC). Conditions on the system parameters for the coexistence of two CLCs and the occurrence of a saddle-node bifurcation of these CLCs are provided. Furthermore, the local asymptotic stability of the pseudo-equilibrium point is analyzed and applications in discontinuous control systems are presented.

Made available in DSpace on 2020-12-10T17:36:52Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-07-09 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) PRONEX/FAPEG Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) The aim of this paper is to study the qualitative dynamics of a piecewise smooth system modeling the intermittent treatment of the human immunodeficiency virus. Typical singularities and closed orbits are observable, and we quantitatively explore the dynamics around those singularities and closed orbits. Moreover, we conclude that this protocol always will be successful since the trajectory passing through any initial condition converges to one of these distinguished orbits. Our formal mathematical results corroborate the real-world observation, where the virus is not eliminated, but the number of infected cells is controlled around a specific value. Univ Sao Paulo, Fac Filosofia Ciencias & Letras Ribeirao Preto, Av Bandeirantes 3900, BR-14040901 Ribeirao Preto, SP, Brazil Univ Fed Goias, Inst Matemat & Estatist, Campus Samambaia, BR-74690900 Goiania, Go, Brazil Univ Estadual Paulista, Inst Biociencias Letras & Ciencias Exatas, Rua Cristovao Colombo 2265, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil Univ Estadual Paulista, Inst Biociencias Letras & Ciencias Exatas, Rua Cristovao Colombo 2265, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil CAPES: 88881.0 68462/2014-01 PRONEX/FAPEG: 2017/10267000-508 CAPES: 001 CNPq: 304809/2017-9 : 2017/00883-0 : 2019/10450-0

In this paper, Hopf and homoclinic bifurcations that occur in the sliding vector field of switching systems in R 3 are studied. In particular, a dc–dc boost converter with sliding mode control and washout filter is analyzed. This device is modeled as a three-dimensional Filippov system, characterized by the existence of sliding movement and restricted to the switching manifold. The operating point of the converter is a stable pseudo-equilibrium and it undergoes a subcritical Hopf bifurcation. Such a bifurcation occurs in the sliding vector field and creates, in this field, an unstable limit cycle. The limit cycle is connected to the switching manifold and disappears when it touches the visible–invisible two-fold point, resulting in a homoclinic loop which itself closes in this two-fold point. The study of these dynamic phenomena that can be found in different power electronic circuits controlled by sliding mode control strategies are relevant from the viewpoint of the global stability and robustness of the control design.