Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems

This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. Several such codimension-one events have recently been identified, causing for example period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighbourhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the the grazing cycle is itself degenerate (e.g. non-hyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that have discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

Dynamics of symmetric dynamical systems with delayed switching

Abstract: We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value, so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity-induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric periodic orbits in systems with full reflection symmetry, a symmetry that is prevalent in applications. We derive an implicit expression for the Poincaré map near the colliding periodic orbit. The Poincaré map is piecewise smooth, finite-dimensional, and changes the dimension of its image at the collision. In the second part of the paper we apply this general result to the class of unstable linear single-degree-of-freedom oscillators where we detect and continue numerically collisions of invariant tori. Moreover, we observe that attracting closed invariant polygons emerge at the torus collision

Transmission Properties of FeCl₃-Intercalated Graphene and WS₂ Thin Films for Terahertz Time-Domain Spectroscopy Applications

This is the final version. Available on open access from Springer via the DOI in this recordTime-resolved terahertz spectroscopy has become a common method both for fundamental and applied studies focused on improving the quality of human life. However, the issue of finding materials applicable in these systems is still relevant. One of the appropriate solutions is 2D materials. Here, we demonstrate the transmission properties of unique graphene-based structures with iron trichloride FeCl3 dopant on glass, sapphire and Kapton polyimide film substrates that previously were not investigated in the framework of the above-described problems in near infrared and THz ranges. We also show properties of a thin tungsten disulfide WS2 film fabricated from liquid crystal solutions transferred to a polyimide and polyethylene terephthalate substrates. The introduction of impurities, the selection of structural dimensions and the use of an appropriate substrate for modified 2D layered materials allow to control the transmission of samples for both the terahertz and infrared ranges, which can be used for creation of effective modulators and components for THz spectroscopy systems.Engineering and Physical Sciences Research Council (EPSRC)Government of the Russian Federatio

An Alternate STAT6-Independent Pathway Promotes Eosinophil Influx into Blood during Allergic Airway Inflammation

Enhanced eosinophil responses have critical roles in the development of allergic diseases. IL-5 regulates the maturation, migration and survival of eosinophils, and IL-5 and eotaxins mediate the trafficking and activation of eosinophils in inflamed tissues. CD4⁺ Th2 cells are the main producers of IL-5 and other cells such as NK also release this cytokine. Although multiple signalling pathways may be involved, STAT6 critically regulates the differentiation and cytokine production of Th2 cells and the expression of eotaxins. Nevertheless, the mechanisms that mediate different parts of the eosinophilic inflammatory process in different tissues in allergic airway diseases remain unclear. Furthermore, the mechanisms at play may vary depending on the context of inflammation and microenvironment of the involved tissues. We employed a model of allergic airway disease in wild type and STAT6-deficient mice to explore the roles of STAT6 and IL-5 in the development of eosinophilic inflammation in this context. Quantitative PCR and ELISA were used to examine IL-5, eotaxins levels in serum and lungs. Eosinophils in lung, peripheral blood and bone marrow were characterized by morphological properties. CD4⁺ T cell and NK cells were identified by flow cytometry. Antibodies were used to deplete CD4⁺ and NK cells. We showed that STAT6 is indispensible for eosinophilic lung inflammation and the induction of eotaxin-1 and -2 during allergic airway inflammation. In the absence of these chemokines eosinophils are not attracted into lung and accumulate in peripheral blood. We also demonstrate the existence of an alternate STAT6-independent pathway of IL-5 production by CD4⁺ and NK cells that mediates the development of eosinophils in bone marrow and their subsequent movement into the circulation

Binary and Millisecond Pulsars at the New Millennium

We review the properties and applications of binary and millisecond pulsars. Our knowledge of these exciting objects has greatly increased in recent years, mainly due to successful surveys which have brought the known pulsar population to over 1300. There are now 56 binary and millisecond pulsars in the Galactic disk and a further 47 in globular clusters. This review is concerned primarily with the results and spin-offs from these surveys which are of particular interest to the relativity community.Comment: 59 pages, 26 figures, 5 tables. Accepted for publication in Living Reviews in Relativity (http://www.livingreviews.org

The Cosmological Constant

This is a review of the physics and cosmology of the cosmological constant. Focusing on recent developments, I present a pedagogical overview of cosmology in the presence of a cosmological constant, observational constraints on its magnitude, and the physics of a small (and potentially nonzero) vacuum energy.Comment: 50 pages. Submitted to Living Reviews in Relativity (http://www.livingreviews.org/), December 199