A unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems

Piecewise smooth dynamical systems make use of discontinuities to model switching between regions of smooth evolution. This introduces an ambiguity in prescribing dynamics at the discontinuity: should the dynamics be given by a limiting value on one side or other of the discontinuity, or a member of some set containing those values? One way to remove the ambiguity is to regularize the discontinuity, the most common being either to smooth it out, or to introduce a hysteresis between switching in one direction or the other across it. Here we show that the two can in general lead to qualitatively different dynamical outcomes. We then define a higher dimensional model with both smoothing and hysteresis, and study the competing limits in which hysteretic or smoothing effects dominate the behaviour, only the former of which correspond to Filippov’s standard ‘sliding modes’.Peer ReviewedPostprint (author's final draft

Chaos in the hysteretic grazing-sliding codimension-one saddle-node bifurcation of piecewise dynamical systems

We present two ways of regularizing a parameter family of piecewise smooth dynamical systems undergoing a grazing- sliding bifurcation. We use the Sotomayor-Teixeira regularization and prove that the bifurcation is a saddle-node (see [ ? ]). Then we perform a hysteretic regularization. However, in spite that the two regularization will give the same dynamics in the sliding modes (see [ ? ]), when a tangency appears, so is in the case of grazing-sliding, the hysteretic process generate chaotic dynamics. Finally, we smooth the hysteresis by embedding the system in a higher dimension. Now the discontinuous control variable u is also a continuous time dependent variable although a fast-fast one. We then encounter loop feedback chaotic behaviourPostprint (author's final draft

Regularization around a generic codimension one fold-fold singularity

This paper is devoted to study the generic fold-fold singularity of Filippov systems on the plane, its unfoldings and its Sotomayor–Teixeira regularization. We work with general Filippov systems and provide the bifurcation diagrams of the fold-fold singularity and their unfoldings, proving that, under some generic conditions, is a codimension one embedded submanifold of the set of all Filippov systems. The regularization of this singularity is studied and its bifurcation diagram is shown. In the visible–invisible case, the use of geometric singular perturbation theory has been useful to give the complete diagram of the unfolding, specially the appearance and disappearance of periodic orbits that are not present in the Filippov vector field. In the case of a linear regularization, we prove that the regularized system is equivalent to a general slow-fast system studied by Krupa and SzmolyanPeer ReviewedPostprint (published version

Traveling waves in a model for cortical spreading depolarization with slow-fast dynamics

Cortical spreading depression and spreading depolarization (CSD) are waves of neuronal depolarization that spread across the cortex, leading to a temporary saturation of brain activity. They are associated with various brain disorders such as migraine and ischemia. We consider a reduced version of a biophysical model of a neuron–astrocyte network for the initiation and propagation of CSD waves [Huguet et al., Biophys. J. 111(2), 452–462, 2016], consisting of reaction-diffusion equations. The reduced model considers only the dynamics of the neuronal and astrocytic membrane potentials and the extracellular potassium concentration, capturing the instigation process implicated in such waves. We present a computational and mathematical framework based on the parameterization method and singular perturbation theory to provide semi-analytical results on the existence of a wave solution and to compute it jointly with its velocity of propagation. The traveling wave solution can be seen as a heteroclinic connection of an associated system of ordinary differential equations with a slow–fast dynamics. The presence of distinct time scales within the system introduces numerical instabilities, which we successfully address through the identification of significant invariant manifolds and the implementation of the parameterization method. Our results provide a methodology that allows to identify efficiently and accurately the mechanisms responsible for the initiation of these waves and the wave propagation velocity.Work produced with support of the grant PID-2021-122954NB-100 funded by MCIN/AEI/ 10.13039/501100011033 and “ERDF: A way of making Europe.” T.M.S and G.H acknowledge the Maria de Maeztu Award for Centers and Units of Excellence in R&D (No. CEX2020-001084-M). T.M.S. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2019. We also acknowledge the use of the UPC Dynamical Systems group’s cluster for research computing (https://dynamicalsystems.upc.edu/en/computing/)Peer ReviewedPostprint (published version

Mycoplasma pneumoniae causing nervous system lesion and SIADH in the absence of pneumonia

A patient was admitted for fever and acute respiratory failure (ARF), rapidly progressive tetraparesis, delirium, behavioral abnormalities, and diplopia. Leukocytosis and a rise in C-reactive protein were present. A syndrome of inappropriate anti-diuretic hormone secretion (SIADH) was also diagnosed. Lumbar puncture yielded colorless CFS with mononuclear pleocytosis and protein rise. Electrodiagnosis revealed demyelinating polyneuropathy and axonal degeneration. Serum IgG and IgM for mycoplasma pneumoniae (MP) was consistent with acute infection, and erythromycin was started with rapid resolution of symptoms. Contrarily to most reports, an associated respiratory disease was not present and SIADH in association with MP has been reported only once, in a patient without direct central nervous system (CNS) involvement. Differential diagnosis and possible pathogenic mechanisms are discussed

Tetrasubstituted Imidazolium Salts as Potent Antiparasitic Agents against African and American Trypanosomiases.

Imidazolium salts are privileged compounds in organic chemistry, and have valuable biological properties. Recent studies show that symmetric imidazolium salts with bulky moieties can display antiparasitic activity against T. cruzi. After developing a facile methodology for the synthesis of tetrasubstituted imidazolium salts from propargylamines and isocyanides, we screened a small library of these adducts against the causative agents of African and American trypanosomiases. These compounds display nanomolar activity against T. brucei and low (or sub) micromolar activity against T. cruzi, with excellent selectivity indexes and favorable molecular properties, thereby emerging as promising hits for the treatment of Chagas disease and sleeping sickness

The regularized visible fold revisited

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $\epsilon\rightarrow 0$. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit $\epsilon\rightarrow 0$, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law