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Papers by KAMILA DA SILVA ANDRADE
International Journal of Bifurcation and Chaos
In a smooth dynamical system, a homoclinic connection is an orbit connecting a saddle equilibrium... more In a smooth dynamical system, a homoclinic connection is an orbit connecting a saddle equilibrium to itself. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interactions with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection to cross or slide along the discontinuity. Even the simplest case of connection to a regular saddle, which hits a discontinuity as a parameter is varied, is surprisingly complex. In this paper, we construct bifurcation diagrams for nonresonant saddles in the plane, unfolding the homoclinic connection to a boundary saddle in a nonsmooth dynamical system. As an application, we exhibit such diagrams for a model of a forced pendulum.
Neste trabalho, estuda-se ciclos que ocorrem tipicamente em campos vetoriais descontínuos, planar... more Neste trabalho, estuda-se ciclos que ocorrem tipicamente em campos vetoriais descontínuos, planares definidos em duas zonas, = (,), com variedade de descontinuidade dada pela imagem inversa do 0 por uma função suave h, definida no plano e assumindo valores reais, para a qual 0 é um valor regular. Primeiramente, mostra-se que, se e são campos vetoriais analíticos e é um policiclo de , então, genericamente, não existem ciclos limite se acumulando em. Depois disso, o objetivo é estudar bifurcações de ciclos típicos contendo um ponto do tipo sela-regular. Mais especificamente, considera-se ciclos compostos por um segmento de órbita regular de , que cruza a variedade de descontinuidade transversalmente, e um ponto do tipo sela-regular resultando numa conexão quase-homoclínica. São apresentados diagramas de bifurcação para o caso onde o raio de hiperbolicidade do ponto de sela é um número irracional, o caso onde o raio de hiperbolicidade da sela é um número racional é ilustrado com alguns modelos. Finalmente, dois modelos comuns em aplicações e que apresentam tal ciclo são estudados por meio de cálculos numéricos.
arXiv: Dynamical Systems, 2019
In this paper, we are concerned about the qualitative behaviour of planar Filippov systems around... more In this paper, we are concerned about the qualitative behaviour of planar Filippov systems around some typical minimal sets, namely, polycycles. In the smooth context, a polycycle is a simple closed curve composed by a collection of singularities and regular orbits, inducing a first return map. Here, this concept is extended to Filippov systems by allowing typical singularities lying on the switching manifold. Our main goal consists in developing a method to investigate the unfolding of polycycles in Filippov systems. In addition, we applied this method to describe bifurcation diagrams of Filippov systems around certain polycycles.
arXiv: Dynamical Systems, 2017
In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equ... more In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interaction with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection allowed to cross or slide along the discontinuity. Even the simplest case, that of connection to a regular saddle that hits a discontinuity as a parameter is varied, is surprisingly complex. Bifurcation diagrams are presented here for non-resonant saddles in the plane, including an example in a forced pendulum.
The main objective of this paper is to study bifurcations of a vector field in a neighborhood of ... more The main objective of this paper is to study bifurcations of a vector field in a neighborhood of a cycle having a homoclinic-like connection at a saddle-regular point. In order to perform such a study it is necessary to analyze how the cycle can be broken, in this way the approach is to look separately at local bifurcations and at the structure of the first return map defined near the cycle.
Journal of Differential Equations, 2021
In this paper, we are interested in providing lower estimations for the maximum number of limit c... more In this paper, we are interested in providing lower estimations for the maximum number of limit cycles H(n) that planar piecewise linear differential systems with two zones separated by the curve y = x n can have, where n is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that H(2) ≥ 4, H(3) ≥ 8, H(n) ≥ 7, for n ≥ 4 even, and H(n) ≥ 9, for n ≥ 5 odd. This improves all the previous results for n ≥ 2. Our analysis is mainly based on some recent results about Chebyshev systems with positive accuracy and Melnikov Theory, which will be developed at any order for a class of nonsmooth differential systems with nonlinear switching manifold.
Journal of Differential Equations, 2019
This paper attempts to present a version of Dulac's problem for piecewise analytic vector fields ... more This paper attempts to present a version of Dulac's problem for piecewise analytic vector fields that states conditions for the birth of the number of limit cycles around certain minimal sets. A suitable model-theoretic structure is introduced under which a qualitative investigation of the problem is settled.
International Journal of Bifurcation and Chaos
In a smooth dynamical system, a homoclinic connection is an orbit connecting a saddle equilibrium... more In a smooth dynamical system, a homoclinic connection is an orbit connecting a saddle equilibrium to itself. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interactions with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection to cross or slide along the discontinuity. Even the simplest case of connection to a regular saddle, which hits a discontinuity as a parameter is varied, is surprisingly complex. In this paper, we construct bifurcation diagrams for nonresonant saddles in the plane, unfolding the homoclinic connection to a boundary saddle in a nonsmooth dynamical system. As an application, we exhibit such diagrams for a model of a forced pendulum.
Neste trabalho, estuda-se ciclos que ocorrem tipicamente em campos vetoriais descontínuos, planar... more Neste trabalho, estuda-se ciclos que ocorrem tipicamente em campos vetoriais descontínuos, planares definidos em duas zonas, = (,), com variedade de descontinuidade dada pela imagem inversa do 0 por uma função suave h, definida no plano e assumindo valores reais, para a qual 0 é um valor regular. Primeiramente, mostra-se que, se e são campos vetoriais analíticos e é um policiclo de , então, genericamente, não existem ciclos limite se acumulando em. Depois disso, o objetivo é estudar bifurcações de ciclos típicos contendo um ponto do tipo sela-regular. Mais especificamente, considera-se ciclos compostos por um segmento de órbita regular de , que cruza a variedade de descontinuidade transversalmente, e um ponto do tipo sela-regular resultando numa conexão quase-homoclínica. São apresentados diagramas de bifurcação para o caso onde o raio de hiperbolicidade do ponto de sela é um número irracional, o caso onde o raio de hiperbolicidade da sela é um número racional é ilustrado com alguns modelos. Finalmente, dois modelos comuns em aplicações e que apresentam tal ciclo são estudados por meio de cálculos numéricos.
arXiv: Dynamical Systems, 2019
In this paper, we are concerned about the qualitative behaviour of planar Filippov systems around... more In this paper, we are concerned about the qualitative behaviour of planar Filippov systems around some typical minimal sets, namely, polycycles. In the smooth context, a polycycle is a simple closed curve composed by a collection of singularities and regular orbits, inducing a first return map. Here, this concept is extended to Filippov systems by allowing typical singularities lying on the switching manifold. Our main goal consists in developing a method to investigate the unfolding of polycycles in Filippov systems. In addition, we applied this method to describe bifurcation diagrams of Filippov systems around certain polycycles.
arXiv: Dynamical Systems, 2017
In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equ... more In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interaction with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection allowed to cross or slide along the discontinuity. Even the simplest case, that of connection to a regular saddle that hits a discontinuity as a parameter is varied, is surprisingly complex. Bifurcation diagrams are presented here for non-resonant saddles in the plane, including an example in a forced pendulum.
The main objective of this paper is to study bifurcations of a vector field in a neighborhood of ... more The main objective of this paper is to study bifurcations of a vector field in a neighborhood of a cycle having a homoclinic-like connection at a saddle-regular point. In order to perform such a study it is necessary to analyze how the cycle can be broken, in this way the approach is to look separately at local bifurcations and at the structure of the first return map defined near the cycle.
Journal of Differential Equations, 2021
In this paper, we are interested in providing lower estimations for the maximum number of limit c... more In this paper, we are interested in providing lower estimations for the maximum number of limit cycles H(n) that planar piecewise linear differential systems with two zones separated by the curve y = x n can have, where n is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that H(2) ≥ 4, H(3) ≥ 8, H(n) ≥ 7, for n ≥ 4 even, and H(n) ≥ 9, for n ≥ 5 odd. This improves all the previous results for n ≥ 2. Our analysis is mainly based on some recent results about Chebyshev systems with positive accuracy and Melnikov Theory, which will be developed at any order for a class of nonsmooth differential systems with nonlinear switching manifold.
Journal of Differential Equations, 2019
This paper attempts to present a version of Dulac's problem for piecewise analytic vector fields ... more This paper attempts to present a version of Dulac's problem for piecewise analytic vector fields that states conditions for the birth of the number of limit cycles around certain minimal sets. A suitable model-theoretic structure is introduced under which a qualitative investigation of the problem is settled.