A singular approach to discontinuous vector fields on the plane

Buzzi, Claudio A.; Silva, Paulo Ricardo da; Teixeira, Marco Antônio

doi:10.1016/j.jde.2006.08.017

Cited by 78 publications

(81 citation statements)

References 8 publications

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“…Discontinuous systems have been studied in [35,36] using a smooth approximation of the discontinuity, followed by the use of singular perturbation theory to obtain the dynamics on a slow manifold. If this approach is followed for system (1), then the equilibrium set is represented by an equilibrium point on the slow manifold.…”

Section: Discussionmentioning

confidence: 99%

Bifurcations of equilibrium sets in mechanical systems with dry friction

Biemond

Wouw

Nijmeijer

2012

Physica D: Nonlinear Phenomena

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Section: Discussionmentioning

confidence: 99%

Bifurcations of equilibrium sets in mechanical systems with dry friction

Biemond

Wouw

Nijmeijer

2012

Physica D: Nonlinear Phenomena

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“…The following definitions of all types of equilibria of Filippov system (2.3) are necessary throughout the paper [19,26,31,32].…”

Section: Sliding Mode Dynamics and Existence Of The Equilibriamentioning

confidence: 99%

Modelling the regulatory system for diabetes mellitus with a threshold window

Tang

Cheke

2015

Communications in Nonlinear Science and Numerical Simulation

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Please cite this article as: Yang, J., Tang, S., Cheke, R.A., Modelling the regulatory system for diabetes mellitus with a threshold window, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http:// dx.doi.org/10.1016/j.cnsns. 2014.08.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.Modelling the regulatory system for diabetes mellitus with a threshold window AbstractPiecewise (or non-smooth) glucose-insulin models with threshold windows for type 1 and type 2 diabetes mellitus are proposed and analyzed with a view to improving understanding of the glucose-insulin regulatory system. For glucose-insulin models with a single threshold, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed. Then the relations between regular equilibria and a pseudo-equilibrium are studied. Furthermore, the sufficient and necessary conditions for the global stability of regular equilibria and the pseudo-equilibrium are provided by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Sliding bifurcations related to boundary node bifurcations were investigated with theoretical and numerical techniques, and insulin clinical therapies are discussed. For glucose-insulin models with a threshold window, the effects of glucose thresholds or the widths of threshold windows on the durations of insulin therapy and glucose infusion were addressed. The duration of the effects of an insulin injection is sensitive to the variation of thresholds. Our results indicate that blood glucose level can be maintained within a normal range using piecewise glucose-insulin models with a single threshold or a threshold window. Moreover, our findings suggest that it is critical to individualize insulin therapy for each patient separately, based on initial blood glucose levels.

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“…It is tangent to Σ and defined at q ∈ Σ S by X S (q) = m − q with m being the point where the segment joining q + X + (q) and q + X − (q) is tangent to Σ ( see [3,10,11,12] for more details and related topics).…”

Section: Non-smooth Dynamical Systemsmentioning

confidence: 99%

Synchronization and Non-Smooth Dynamical Systems

2012

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Abstract. In this article we establish an interaction between nonsmooth systems, geometric singular perturbation theory and synchronization phenomena. We find conditions for a non-smooth vector fields be locally synchronized. Moreover its regularization provide a singular perturbation problem with attracting critical manifold. We also state a result about the synchronization which occurs in the regularization of the fold-fold case. We restrict ourselves to the 3-dimensional systems ( = 3) and consider the case known as a T-singularity.

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A singular approach to discontinuous vector fields on the plane

Cited by 78 publications

References 8 publications

Bifurcations of equilibrium sets in mechanical systems with dry friction

Bifurcations of equilibrium sets in mechanical systems with dry friction

Modelling the regulatory system for diabetes mellitus with a threshold window

Synchronization and Non-Smooth Dynamical Systems

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