Joint Meeting Spai-Brazil in Mathematics

Presentation's date: 2018-12-11

Journal of differential equations

Vol. 265, num. 5, p. 1761-1838

DOI: 10.1016/j.jde.2018.04.047

Date of publication: 2018-09-05

Abstract:

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 Szmolyan]]>

Communications in nonlinear science and numerical simulation

Vol. 50, p. 142-168

DOI: 10.1016/j.cnsns.2017.02.014

Date of publication: 2017-09-01

Abstract:

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’.]]>

Congreso de Ecuaciones Diferenciales y Aplicaciones y Congreso de Matemàtica Aplicada

p. 73-77

Presentation's date: 2017-06-28

Abstract:

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 behaviour]]>

Discrete and continuous dynamical systems. Series A

Vol. 36, num. 7, p. 3545-3601

DOI: 10.3934/dcds.2016.36.3545

Date of publication: 2016-07-01

Abstract:

In this paper we study the Sotomayor-Teixeira regularization of a general visible fold singularity of a planar Filippov system. Extending Geometric Fenichel Theory beyond the fold with asymptotic methods, we determine the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is applied to the regularization of global sliding bifurcations as the Grazing-Sliding of periodic orbits and the Sliding Homoclinic to a Saddle, as well as to some classical problems in dry friction.; Roughly speaking, we see that locally, and also globally, the regularization of the bifurcations preserve the topological features of the sliding ones.]]>

Abstract:

This project belongs to the area of Dynamical Systems and Applications, and it is the natural continuation of projects MTM2006-00478, MTM2009-06973 and MTM2012-31714, focusing on the local study and, mainly, the global study of continuous and discrete dynamical systems using analytical and numerical tools. The Dynamical Systems Group at the UPC is broad and interdisciplinary, combining experienced researchers with young researchers. This high research potential, coupled with a solid theoretical base, makes it possible to cover a wide set of classic and new problems in Dynamical Systems. The goal of the project consists on keep our leadership in the areas of Arnold Diffusion, splitting of separatrices and Astrodynamics, as well as advance in the study of bifurcations, computation of invariant objects and integrability. But we also want to strengthen the application of Dynamical Systems tools to problems in Astrodynamics, Celestial Mechanics, Chemistry and, as was most recently done with great success, in infinite dimensional Dynamical Systems and Neuroscience. In order to obtain numerical results not only in some theoretical problems, like exponentially small splitting of separatrices, but also in applications to Astrodynamics, Celestial Mechanics or Neuroscience, a highly computational approach is necessary, and very often it is only possible through parallel computation (the group owns and maintains a HPC cluster). The group already has a widely recognized expertise and prestige in these fields, and we plan to continue working on them in the following years Finally, it is important to stress that the group seeks a balance between working on issues where it is already expert and internationally recognized, and do it in new and more challenging problems for the group, such as infinite dimensional Dynamical Systems and Neuroscience. Our goals for the next three years are the following: A. Arnold Diffusion A.1. A-priori unstable Systems. A.2. A-priori stable Systems. A.3. Applications. B. Exponentially small phenomena B.1. Splitting of separatrices to low dimensional invariant objects. B.2. Splitting of separatrices in the Hopf-zero singularity. B.3. Using Melnikov potential in the problem of splitting of separatrices. B.4. Billiards next to the boundary. C. Integrability C.1. Planar polinomial vector fields. C.2. Algebraic integrable systems. D. Invariant objects and their bifurcations D.1. Invariant manifold of parabolic objects. D.2. Filippov Systems. D.3.Planar reversible diffeomorphisms. D.4. Planar vector fields. D.5. Quasi- periodic dynamics. D.6. Applications. E. Infinite Dimensional Dynamical Systems. E.1. Growth of Sobolev norms in Hamiltonian PDEs. E.2.Singular problems in PDEs. F. Astrodynamics F.1. Generation of natural and artificial trajectories. F.2. Artificial satellite formation. F.3. Analysis of orbits in the three body problem. G. Neuroscience and Biomedical applications G.1. Dynamics of models for single neurons and neural populations. G.2. Biomedical applications.]]>

SIAM Conference on Applications of Dynamical Systems

Presentation's date: 2015-05-17

ICMC Summer Meeting on Differential Equations

p. 71

Presentation's date: 2014-02-05

Date: 2014-02

Abstract:

In this paper we study the Sotomayor-Teixeira regularization of a general visible fold singularity of a Filippov system. Extending Geometric Fenichel Theory beyond the fold with asymptotic methods, we determine there the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is applied to the regularization of some global sliding bifurcations as the Grazing-Sliding of periodic orbits or the Sliding Homoclinic to a Saddle, as well as to some classical problems in dry friction. Roughly speaking, we see that locally, and also globally, the regularization of the bifurcations preserve the topological features of the sliding ones]]>

SIAM Conference on Applications of Dynamical Systems

p. 154

Presentation's date: 2013-05-21

Abstract:

We combine geometric singular perturbation theory with matching perturbation methods to study the local map of near-grazing solutions at the boundary between crossing and sliding dynamics, in the smoothing of a Filippov-type nonsmooth dynamical system. We show that the Lipchitz constant is of order O(ea), a > 0, in terms of the singular perturbation parameter e. The value of a depends on the smoothness of the function regularizing the discontinuity. The persistence of periodic orbits is discussed.]]>

Date of publication: 2002-09

SIAM journal on mathematical analysis

Vol. 29, num. 6, p. 1335-1360

Date of publication: 1998-11

Date: 1997-05

Date: 1996-12

Congreso de Ecuaciones Diferenciales y Aplicaciones. Congreso de Matemática Aplicada

Presentation's date: 1995-09-18

Congreso de Ecuaciones Diferenciales y Aplicaciones. Congreso de Matemática Aplicada

p. 167-168

Date of publication: 1992-01

Date of publication: 1991-09