Closed poly-trajectories and Poincaré index of non-smooth vector fields on the plane

Buzzi, Claudio A.; Carvalho, Tiago; Silva, P. R.

doi:10.1007/s10883-013-9169-4

Cited by 28 publications

(41 citation statements)

References 9 publications

(22 reference statements)

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“…In Subsection 2.3 we prove Lemma 8, i.e., we exhibit the homeomorphism that characterizes the equivalence between any fold−cusp singularity and the standard form given by (5).…”

Section: Consider the Notationmentioning

confidence: 98%

On 3-Parameter Families of Piecewise Smooth Vector Fields in the Plane

Buzzi¹,

Carvalho²,

Teixeira³

2012

SIAM J. Appl. Dyn. Syst.

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Abstract. This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three−parameter families of a class of non−smooth vector fields are studied and the tridimensional bifurcation diagrams are exhibited. Our main results describe the unfolding of the so called f old − cusp singularity by means of the variation of 3 parameters.

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“…In Subsection 2.3 we prove Lemma 8, i.e., we exhibit the homeomorphism that characterizes the equivalence between any fold−cusp singularity and the standard form given by (5).…”

Section: Consider the Notationmentioning

confidence: 98%

On 3-Parameter Families of Piecewise Smooth Vector Fields in the Plane

Buzzi¹,

Carvalho²,

Teixeira³

2012

SIAM J. Appl. Dyn. Syst.

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“…Now we introduce the definitions of several type of equilibria, which are very useful in the following parts [4,5,10,19].…”

Section: Preliminariesmentioning

confidence: 99%

“…When ET < x L 2 , then Ω 2 is a limit cycle of system (2.1), as shown in Figure 4 (a) with ET = 0.1. If ET > x L 2 , then the touching cycle becomes a canard cycle [4], as shown in Figure 4 (c). Sliding crossing bifurcation: If we choose ET as a bifurcation parameter and fix all other parameters as those in Figure 5, then a sliding crossing bifurcation can be observed.…”

Section: Global Sliding Bifurcationsmentioning

confidence: 99%

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Sliding Bifurcation Analysis and Global Dynamics for a Filippov Predator-prey System

Tan¹,

Qin²,

Liu³

et al. 2016

J. Nonlinear Sci. Appl.

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This paper studies a Filippov predator-prey system, where chemical control strategies are proposed and analyzed. Initially, the exact sliding segment and its domains are addressed. Then the existence and stability of the regular, virtual, pseudo-equilibria and tangent points are discussed. It shows that two regular equilibria and a pseudo-equilibrium can coexist. By employing theoretical and numerical techniques several kinds of bifurcations are investigated, such as sliding bifurcations related to the boundary node (focus) bifurcations, touching bifurcations, sliding crossing bifurcation and buckling bifurcations (or sliding switching). Furthermore, it makes comparison of the obtained results with previous studies for the Filippov predator-prey system without control strategies. Some biological implications of our results with respect to pest control are also given.

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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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Closed poly-trajectories and Poincaré index of non-smooth vector fields on the plane

Cited by 28 publications

References 9 publications

On 3-Parameter Families of Piecewise Smooth Vector Fields in the Plane

On 3-Parameter Families of Piecewise Smooth Vector Fields in the Plane

Sliding Bifurcation Analysis and Global Dynamics for a Filippov Predator-prey System

Modelling the regulatory system for diabetes mellitus with a threshold window

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