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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.

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.