arXiv subjects

Durval J. Tonon

Publications and source records attributed to Durval J. Tonon.

4 recordsLinked to original sources

arXiv2024

Sliding mode on tangential sets of Filippov systems

We consider piecewise smooth vector fields $Z=(Z_+, Z_-)$ defined in $\mathbb{R}^n$ where both vector fields are tangent to the switching manifold $Σ$ along a submanifold $M\subset Σ$. We shall see that, under suitable assumptions, Filippov convention gives rise to a unique sliding mode on $M$, governed by what we call the {\it tangential sliding vector field}. Here, we will provide the necessary and sufficient conditions for characterizing such a vector field. Additionally, we prove that the tangential sliding vector field is conjugated to the reduced dynamics of a singular perturbation problem arising from the Sotomayor-Teixeira regularization of $Z$ around $M$. Finally, we analyze several examples where tangential sliding vector fields can be observed, including a model for intermittent treatment of HIV.

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arXiv2021

Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a real one for $y<0$ and a virtual one for $y>0$, and such that the real center is a global center. Then, working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one additional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.

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arXiv2020

Lower bounds for the number of limit cycles in a generalized Rayleigh-Liénard oscillator

In this paper a generalized Rayleigh-Liénard oscillator is consider and lower bounds for the number of limit cycles bifurcating from weak focus equilibria and saddle connections are provided. By assuming some open conditions on the parameters of the considered system the existence of up to twelve limit cycles is provided. More precisely, the approach consists in perform suitable changes in the sign of some specific parameters and apply Poincaré-Bendixson Theorem for assure the existence of limit cycles. In particular, the method for obtaining the limit cycles through the referred approach is explicitly exhibited. The main techniques applied in this study are the Lyapunov constants and the Melnikov method.

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arXiv2013

An asymptotically stable cusp-fold singularity in 3D piecewise smooth vector fields

This paper is concerned with the analysis of a typical singularity of piecewise smooth vector fields on $R^3$ composed by two zones. In our object of study, the cusp-fold singularity, we consider the simultaneous occurrence of a cusp singularity for one vector field and a fold singularity for the other one. We exhibit a normal form that presents one of the most important property searched for in piecewise smooth vector fields: the asymptotical stability.

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