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Papers by durval tonon

arXiv (Cornell University), Nov 21, 2011

This paper treats on the existence of closed orbits around a two-fold singularity of 3D discontin... more This paper treats on the existence of closed orbits around a two-fold singularity of 3D discontinuous systems of the Filippov type in presence of symmetries.

Phippov systems in tridimensional manifolds

Orientador: Marco Antonio TeixeiraTese (doutorado) - Universidade Estadual de Campinas, Instituto... more Orientador: Marco Antonio TeixeiraTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: Neste trabalho sistemas dinâmicos descontínuos em variedades tridimensionais são estudados. Descrevemos uma classe de tais sistemas que são localmente estruturalmente estáveis em uma vizinhança de uma singularidade típica. Exibimos nessa etapa uma sub-família de campos do tipo dobra-dobra que é estruturalmente estável. Introduzimos os conceitos de A e L-estabilidade, que são pequenas generalizações dos conceitos clássicos de estabilidade assintótica e estabilidade no sentido de Lyapunov, respectivamente. Através de formas normais para as famílias de campos descontínuos de codimensão zero e um, exibimos os subconjuntos de sistemas descontínuos que são A e L-estáveis em uma vizinhança da origem. Destacamos um dos principais objetos de estudo desse trabalho: a singularidade dobra-dobra caso elíptico (T-singularidade). Discutimos alg...

arXiv (Cornell University), Jun 6, 2013

This paper is concerned with the analysis of a typical singularity of piecewise smooth vector fie... more 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.

Dynamical Systems-an International Journal, Nov 2, 2018

This paper studies the global dynamics of piecewise smooth differential equations defined in the ... more This paper studies the global dynamics of piecewise smooth differential equations defined in the two-dimensional torus and sphere in the case when the switching manifold breaks the manifold into two connected components. Over the switching manifold, we consider the Filippov's convention for discontinuous differential equations. The study of piecewise smooth dynamical systems over torus and sphere is common for maps and up to where we know this is the first characterization for piecewise smooth flows arising from solutions of differential equations. We provide conditions under generic families of piecewise smooth equations to get periodic and dense trajectories. Considering these generic families of piecewise differential equations, we prove that a non-deterministic chaotic behaviour appears. Global bifurcations are also classified.

Lower bounds for the number of limit cycles in a generalised Rayleigh–Liénard oscillator

Nonlinearity, Jun 22, 2022

In this paper a generalised Rayleigh–Liénard oscillator is consider and lower bounds for the numb... more In this paper a generalised 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 algorithm 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. The obtained results contemplate the simultaneity of limit cycles of small amplitude and medium amplitude, the former emerging from a weak focus equilibrium and the latter from homoclinic or heteroclinic saddle connections.

Journal of Mathematical Physics, Feb 1, 2017

In this paper we study the existence of periodic symmetric orbits of the 3-body problem when each... more In this paper we study the existence of periodic symmetric orbits of the 3-body problem when each body possess mass and an electric charge. The main technique applied in this study is the continuation method of Poincaré.

Journal of Mathematical Analysis and Applications, May 1, 2017

When the first average function is non-zero we provide an upper bound for the maximum number of l... more When the first average function is non-zero we provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of the centerẋ = −y((x 2 + y 2)/2) m andẏ = x((x 2 + y 2)/2) m with m ≥ 1, when we perturb it inside a class of discontinuous piecewise polynomial vector fields of degree n with k pieces. The positive integers m, n and k are arbitrary. The main tool used for proving our results is the averaging theory for discontinuous piecewise vector fields.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

International Journal of Bifurcation and Chaos, Jul 1, 2014

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a sce... more In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

Piecewise Smooth Reversible Dynamical Systems at a Two-Fold Singularity

International Journal of Bifurcation and Chaos, Aug 1, 2012

Global Analysis of the Dynamics of a Piecewise Linear Vector Field Model for Prostate Cancer Treatment

Journal of Dynamical and Control Systems, Oct 29, 2021

RSME Springer series, 2020

In this paper aspects of local asymptotic and Lyapunov stability of 3D piecewise smooth vector fi... more In this paper aspects of local asymptotic and Lyapunov stability of 3D piecewise smooth vector fields are studied.

Letters in Mathematical Physics, Feb 5, 2018

We analyse the existence of periodic symmetric orbits of the classical helium atom. The results o... more We analyse the existence of periodic symmetric orbits of the classical helium atom. The results obtained shows that there exists six families of periodic orbits that can be prolonged from a continuum of periodic symmetric orbits. The main technique applied in this study is the continuation method of Poincaré.

Discrete and Continuous Dynamical Systems-series B, Nov 1, 2015

Using the averaging theory we study the periodic solutions and their linear stability of the 3-di... more Using the averaging theory we study the periodic solutions and their linear stability of the 3-dimensional chaotic quadratic polynomial differential systems without equilibria studied in [3]. All these differential systems depend only on one-parameter.

International Journal of Bifurcation and Chaos, Sep 15, 2020

We characterize the families of periodic orbits of two discontinuous piecewise differential syste... more We characterize the families of periodic orbits of two discontinuous piecewise differential systems in R 3 separated by a plane using their first integrals. One of these discontinuous piecewise differential systems is formed by linear differential systems, and the other by nonlinear differential systems.

arXiv (Cornell University), Nov 24, 2021

We consider piecewise smooth vector fields Z = (Z + , Z −) defined in R n where both vector field... more We consider piecewise smooth vector fields Z = (Z + , Z −) defined in R n where both vector fields are tangent to the switching manifold Σ along a manifold M. Our main purpose is to study the existence of an invariant vector field defined on M , that we call tangential sliding vector field. We provide necessary and sufficient conditions under Z to characterize the existence of this vector field. Considering a regularization process, we proved that this tangential sliding vector field is topological equivalent to an invariant dynamics under the slow manifold. The results are applied to study a Filippov model for intermittent treatment of HIV.

arXiv (Cornell University), Dec 27, 2020

In this paper a generalized Rayleigh-Liénard oscillator is consider and lower bounds for the numb... more 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. 2010 Mathematics Subject Classification. 34C23 and 34C25 and 37G15. Key words and phrases. Rayleigh Liénard oscillator and limit cycles and Lyapunov constants and Melnikov function.

arXiv (Cornell University), Nov 21, 2011

Our object of study is non smooth vector fields on R2\R^2R2. We apply the techniques of geometric s... more Our object of study is non smooth vector fields on R2\R^2R2. We apply the techniques of geometric singular perturbations in non smooth vector fields after regularization and a blow$-$up. In this way we are able to bring out some results that bridge the space between non$-$smooth dynamical systems presenting typical singularities and singularly perturbed smooth systems.

Two-fold symmetric singularity

arXiv (Cornell University), Nov 21, 2011

We explore some qualitative dynamics in the neighborhood of the 3−dimensional3-dimensional3−dimensional two-fold symmetri... more We explore some qualitative dynamics in the neighborhood of the 3−dimensional3-dimensional3−dimensional two-fold symmetric singularity. We study the existence of an one-parameter family of regular (pseudo) periodic orbits of such systems near a reversible two-fold singularity.

arXiv (Cornell University), Jan 21, 2016

In this paper we study the global dynamics of piecewise smooth vector fields defined in the two d... more In this paper we study the global dynamics of piecewise smooth vector fields defined in the two dimensional torus and sphere. We provide conditions under these families exhibits periodic and dense trajectories and we describe some global bifurcations. We also study its minimal sets and characterize the chaotic behavior of the piecewise smooth vector fields defined in torus and sphere.

Physica D: Nonlinear Phenomena, May 1, 2017

h i g h l i g h t s • A dc-dc boost converter with sliding mode control and washout filter is stu... more h i g h l i g h t s • A dc-dc boost converter with sliding mode control and washout filter is studied. • A Hopf bifurcation is observed at the sliding vector field. • A homoclinic loop appears for some choices on the parameters.

arXiv (Cornell University), Nov 21, 2011

This paper treats on the existence of closed orbits around a two-fold singularity of 3D discontin... more This paper treats on the existence of closed orbits around a two-fold singularity of 3D discontinuous systems of the Filippov type in presence of symmetries.

Phippov systems in tridimensional manifolds

Orientador: Marco Antonio TeixeiraTese (doutorado) - Universidade Estadual de Campinas, Instituto... more Orientador: Marco Antonio TeixeiraTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: Neste trabalho sistemas dinâmicos descontínuos em variedades tridimensionais são estudados. Descrevemos uma classe de tais sistemas que são localmente estruturalmente estáveis em uma vizinhança de uma singularidade típica. Exibimos nessa etapa uma sub-família de campos do tipo dobra-dobra que é estruturalmente estável. Introduzimos os conceitos de A e L-estabilidade, que são pequenas generalizações dos conceitos clássicos de estabilidade assintótica e estabilidade no sentido de Lyapunov, respectivamente. Através de formas normais para as famílias de campos descontínuos de codimensão zero e um, exibimos os subconjuntos de sistemas descontínuos que são A e L-estáveis em uma vizinhança da origem. Destacamos um dos principais objetos de estudo desse trabalho: a singularidade dobra-dobra caso elíptico (T-singularidade). Discutimos alg...

arXiv (Cornell University), Jun 6, 2013

This paper is concerned with the analysis of a typical singularity of piecewise smooth vector fie... more 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.

Dynamical Systems-an International Journal, Nov 2, 2018

This paper studies the global dynamics of piecewise smooth differential equations defined in the ... more This paper studies the global dynamics of piecewise smooth differential equations defined in the two-dimensional torus and sphere in the case when the switching manifold breaks the manifold into two connected components. Over the switching manifold, we consider the Filippov's convention for discontinuous differential equations. The study of piecewise smooth dynamical systems over torus and sphere is common for maps and up to where we know this is the first characterization for piecewise smooth flows arising from solutions of differential equations. We provide conditions under generic families of piecewise smooth equations to get periodic and dense trajectories. Considering these generic families of piecewise differential equations, we prove that a non-deterministic chaotic behaviour appears. Global bifurcations are also classified.

Lower bounds for the number of limit cycles in a generalised Rayleigh–Liénard oscillator

Nonlinearity, Jun 22, 2022

In this paper a generalised Rayleigh–Liénard oscillator is consider and lower bounds for the numb... more In this paper a generalised 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 algorithm 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. The obtained results contemplate the simultaneity of limit cycles of small amplitude and medium amplitude, the former emerging from a weak focus equilibrium and the latter from homoclinic or heteroclinic saddle connections.

Journal of Mathematical Physics, Feb 1, 2017

In this paper we study the existence of periodic symmetric orbits of the 3-body problem when each... more In this paper we study the existence of periodic symmetric orbits of the 3-body problem when each body possess mass and an electric charge. The main technique applied in this study is the continuation method of Poincaré.

Journal of Mathematical Analysis and Applications, May 1, 2017

When the first average function is non-zero we provide an upper bound for the maximum number of l... more When the first average function is non-zero we provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of the centerẋ = −y((x 2 + y 2)/2) m andẏ = x((x 2 + y 2)/2) m with m ≥ 1, when we perturb it inside a class of discontinuous piecewise polynomial vector fields of degree n with k pieces. The positive integers m, n and k are arbitrary. The main tool used for proving our results is the averaging theory for discontinuous piecewise vector fields.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

International Journal of Bifurcation and Chaos, Jul 1, 2014

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a sce... more In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

Piecewise Smooth Reversible Dynamical Systems at a Two-Fold Singularity

International Journal of Bifurcation and Chaos, Aug 1, 2012

Global Analysis of the Dynamics of a Piecewise Linear Vector Field Model for Prostate Cancer Treatment

Journal of Dynamical and Control Systems, Oct 29, 2021

RSME Springer series, 2020

In this paper aspects of local asymptotic and Lyapunov stability of 3D piecewise smooth vector fi... more In this paper aspects of local asymptotic and Lyapunov stability of 3D piecewise smooth vector fields are studied.

Letters in Mathematical Physics, Feb 5, 2018

We analyse the existence of periodic symmetric orbits of the classical helium atom. The results o... more We analyse the existence of periodic symmetric orbits of the classical helium atom. The results obtained shows that there exists six families of periodic orbits that can be prolonged from a continuum of periodic symmetric orbits. The main technique applied in this study is the continuation method of Poincaré.

Discrete and Continuous Dynamical Systems-series B, Nov 1, 2015

Using the averaging theory we study the periodic solutions and their linear stability of the 3-di... more Using the averaging theory we study the periodic solutions and their linear stability of the 3-dimensional chaotic quadratic polynomial differential systems without equilibria studied in [3]. All these differential systems depend only on one-parameter.

International Journal of Bifurcation and Chaos, Sep 15, 2020

We characterize the families of periodic orbits of two discontinuous piecewise differential syste... more We characterize the families of periodic orbits of two discontinuous piecewise differential systems in R 3 separated by a plane using their first integrals. One of these discontinuous piecewise differential systems is formed by linear differential systems, and the other by nonlinear differential systems.

arXiv (Cornell University), Nov 24, 2021

We consider piecewise smooth vector fields Z = (Z + , Z −) defined in R n where both vector field... more We consider piecewise smooth vector fields Z = (Z + , Z −) defined in R n where both vector fields are tangent to the switching manifold Σ along a manifold M. Our main purpose is to study the existence of an invariant vector field defined on M , that we call tangential sliding vector field. We provide necessary and sufficient conditions under Z to characterize the existence of this vector field. Considering a regularization process, we proved that this tangential sliding vector field is topological equivalent to an invariant dynamics under the slow manifold. The results are applied to study a Filippov model for intermittent treatment of HIV.

arXiv (Cornell University), Dec 27, 2020

In this paper a generalized Rayleigh-Liénard oscillator is consider and lower bounds for the numb... more 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. 2010 Mathematics Subject Classification. 34C23 and 34C25 and 37G15. Key words and phrases. Rayleigh Liénard oscillator and limit cycles and Lyapunov constants and Melnikov function.

arXiv (Cornell University), Nov 21, 2011

Our object of study is non smooth vector fields on R2\R^2R2. We apply the techniques of geometric s... more Our object of study is non smooth vector fields on R2\R^2R2. We apply the techniques of geometric singular perturbations in non smooth vector fields after regularization and a blow$-$up. In this way we are able to bring out some results that bridge the space between non$-$smooth dynamical systems presenting typical singularities and singularly perturbed smooth systems.

Two-fold symmetric singularity

arXiv (Cornell University), Nov 21, 2011

We explore some qualitative dynamics in the neighborhood of the 3−dimensional3-dimensional3−dimensional two-fold symmetri... more We explore some qualitative dynamics in the neighborhood of the 3−dimensional3-dimensional3−dimensional two-fold symmetric singularity. We study the existence of an one-parameter family of regular (pseudo) periodic orbits of such systems near a reversible two-fold singularity.

arXiv (Cornell University), Jan 21, 2016

In this paper we study the global dynamics of piecewise smooth vector fields defined in the two d... more In this paper we study the global dynamics of piecewise smooth vector fields defined in the two dimensional torus and sphere. We provide conditions under these families exhibits periodic and dense trajectories and we describe some global bifurcations. We also study its minimal sets and characterize the chaotic behavior of the piecewise smooth vector fields defined in torus and sphere.

Physica D: Nonlinear Phenomena, May 1, 2017

h i g h l i g h t s • A dc-dc boost converter with sliding mode control and washout filter is stu... more h i g h l i g h t s • A dc-dc boost converter with sliding mode control and washout filter is studied. • A Hopf bifurcation is observed at the sliding vector field. • A homoclinic loop appears for some choices on the parameters.