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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.

Despite the huge relevance of vaccines for preventing COVID-19, physical isolation and quarantine of infected individuals are still the key strategies to fight against the COVID-19 pandemic. Based on a COVID-19 transmission epidemiological model governed by ordinary differential equations, here we propose an intermittent non-pharmacological protocol to control the fraction of infected individuals. In our approach, unlike what generically happens for numerical simulation models, we provide a global analysis of the model, giving qualitative information about every initial condition. Under some simple hypothesis and variations of parameters, we present some bifurcations and we are able to predict the minimum social distancing effort that do not collapse the health system.

Our starting point is a discontinuous piecewise linear dynamical system in [Formula: see text] with two zones that present a single invisible two-fold straight line and invariant cylinders. We broke the dynamics of this vector field to obtain a new four-parameter piecewise linear vector field with a [Formula: see text]-singularity. For this vector field, we prove that the upper bound of simple crossing limit cycles (simple CLCs) is three, and provide conditions for the existence of 1, 2, or 3 simple CLCs. We also show the occurrence of three distinct bifurcations involving such simple CLCs that are derived from a bifurcation set with two parameters: (i) a Teixeira Singularity bifurcation (TS-bifurcation); (ii) a Fold bifurcation; and (iii) a Cusp bifurcation.

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 electronic circuit of a DC–DC boost power converter under a specific sliding mode control strategy is analyzed. This circuit is modeled as a discontinuous piecewise linear three-dimensional dynamical system, whose state space is divided into two open regions by a plane acting as the switching manifold. This system displays sliding motion confined to the switching manifold and limited by two straight lines where lie the points of tangency, one for each involved linear vector field. Such tangency lines can intersect transversally on a invisible two-fold point known as Teixeira singularity. Our goal is to investigate an interesting bifurcation that occurs on the Teixeira singularity, involving both a pseudo-equilibrium point and a crossing periodic orbit. This bifurcation, named TS-bifurcation, occurs when a pseudo-equilibrium point of the discontinuous piecewise linear system, capable of moving between the attractive and repulsive sliding regions as a result of the change in a parameter, collides with the Teixeira singularity. Simultaneously, a periodic orbit arises from the Teixeira singularity crossing the switching manifold in two points. Apart from the TS-bifurcation characterization in the DC–DC converter, we have numerically detected other non-local phenomena like a saddle-node bifurcation of crossing limit cycles. Experimental results to illustrate the effects of the TS-bifurcation on the dynamics of a power electronic converter are also shown.

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

One of the most interesting typical singularity observed in 3D piecewise-smooth dynamical systems is the so called T-singularity (Teixeira Singularity). Throughout this paper, we study a Fold bifurcation involving two T-singularities that, as far as we know, has not yet been fully addressed in the literature. In this bifurcation both T-singularities collide and then disappear when a system parameter is varied. At the collision point appears a type of non-generic T-singularity. For this study we use a normal form and we apply the standard analysis for such class of systems, based on the analysis of the sliding dynamics (Filippov’s sliding vector field) and the crossing dynamics (first return map) occurring on the switching boundary. We fully describe the unfolding dynamics of the Fold bifurcation under study and also analyze the existence, stability and bifurcations of invariant manifolds in this scenario.

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.

Capacitor voltage-balancing systems are usually applied to power electronic circuits. The main issue in these systems is equalising the voltage of a large number of capacitors connected to a voltage source in serial or parallel arrangements. A particular application of the capacitor voltage-balancing systems is found in modular multilevel converters (MMC). The voltage balancing of the floating capacitors in the submodules of the MMC during its pre-charging operating stage is a key issue since it is a critical task for the correct operating of these converters. The stability analysis of an active voltage balancing strategy for pre-charging MMCs is addressed in this paper. The adopted voltage-balancing strategy consists in adding a resistance to each submodule of the MMC by means of a controlled switch. These switches are being controlled by a sliding mode control algorithm with multiple boundaries (discontinuity surfaces of high co-dimension). These systems are essentially discontinuous piecewise smooth dynamical systems (Filippov systems) commanded by electronic switches. In this paper, the local stability of the voltage balanced system is analytically proven for an arbitrary number of submodules. In addition, a detailed analysis of the global dynamics of this system with two submodules and two switching boundaries sliding mode control is presented. Simulation results obtained on a MMC with 10 submodules are shown to validate the theoretical analysis.

Direct current (DC) microgrids (MGs) are an emergent option to satisfy new demands for power quality and integration of renewable resources in electrical distribution systems. This work addresses the large-signal stability analysis of a DC–DC bidirectional converter (DBC) connected to a storage device in an islanding MG. This converter is responsible for controlling the balance of power (load demand and generation) under constant power loads (CPLs). In order to control the DC bus voltage through a DBC, we propose a robust sliding mode control (SMC) based on a washout filter. Dynamical systems techniques are exploited to assess the quality of this switching control strategy. In this sense, a bifurcation analysis is performed to study the nonlinear stability of a reduced model of this system. The appearance of different bifurcations when load parameters and control gains are changed is studied in detail. In the specific case of Teixeira Singularity (TS) bifurcation, some experimental results are provided, confirming the mathematical predictions. Both a deeper insight in the dynamic behavior of the controlled system and valuable design criteria are obtained.

For a discontinuous piecewise smooth (DPWS) dynamical system in [Formula: see text], whose state space is divided into two open regions by a plane acting as the switching manifold, there generically appear two lines of quadratic tangency, one for each involved vector field. When these two tangency lines have a transversal intersection, such a point is called a two-fold singularity. If furthermore, both tangencies are of invisible type, then the two-fold point is known as a Teixeira singularity (TS). The Teixeira singularity can undergo an interesting bifurcation, namely when a pseudo-equilibrium point crosses the two-fold singularity, passing from the attractive sliding region to the repulsive sliding region (or vice versa) and, simultaneously, a crossing limit cycle (CLC) arises. After deriving carefully a local canonical form, we revisit the previous works regarding this bifurcation thus correcting some detected misconceptions. Furthermore, we provide by means of a more direct approach the critical coefficients characterizing the bifurcation, also giving computational procedures for them. The achieved results are applied to some illustrative examples, within the realm of discontinuous piecewise linear (DPWL) systems. This family acts like a normal form for the bifurcation, since DPWL systems are able to reproduce all the unfolded dynamics. The study of TS-points in electronic DC-DC Boost power converters under a sliding mode control (SMC) strategy is addressed. Apart from being a relevant application, it allows to show the real usefulness of the analysis done.

Este artigo apresenta uma analise de bifurcacoes de um conversor de eletronica potencia boost bidirecional, considerando uma estrategia de controle por modos deslizantes (SMC) para microrredes de corrente continua (cc) em modo ilhado. Esta analise tem por objetivo identificar fenomenos ligados as bifurcacoes dos equilibrios que possam levar o sistema a uma instabilidade ou estabilidade num ponto de equilibrio (ou orbita periodica) nao desejado, e com isso, determinar condicoes sobre os parâmetros para garantir estabilidade e robustez do sistema.

As microredes cc sao uma opcao para novas demandas por qualidade de energia e integracao de fontes renovaveis no sistema eletrico. Numa microrede em modo ilhado, o conversor bidirecional cc-cc (DBC) tem o papel de controlar o balanco de potencia entre geracao e demanda de carga, utilizando a bateria para drenar ou injetar potencia na rede. O conversor bidirecional com controle por modos deslizantes (SMC) e modelado como um sistema dinâmico DPWS. Esta classe de sistemas exibe bifurcacoes tipicas, denominadas bifurcacoes induzidas pela descontinuidade (DIBs), como por exemplo a Bifurcacao do Equilibrio de Fronteira (BEBs), que e foco de estudo deste trabalho.

In this paper the nonlinear dynamics of interconnected power converters in an islanded direct current (DC) microgrid is analyzed. By using a simplified scheme based on two cascaded converters we analyze the dynamical behavior that can arise from the interconnection of these devices on a common DC bus. Furthermore, in order to address the bus voltage control problem, we propose a Sliding Mode Controller for a DC-DC bidirectional power converter to control the DC bus voltage under instantaneous constant power loads. This class of loads introduces a destabilizing nonlinear effect on the converter through an inverse voltage term that can lead to significant oscillations in the DC bus voltage. Simulation results are shown to illustrate the nonlinear analysis.