We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard d... more We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard differential systemṡ x = y,ẏ = (ax + b)y + cx 2 + dx + e, where (x, y) ∈ R 2 are the variables and a, b, c, d, e are real parameters.
Ao término deste trabalho, gostaria de manifestar meus sinceros agradecimentos: A minha mãe pelo ... more Ao término deste trabalho, gostaria de manifestar meus sinceros agradecimentos: A minha mãe pelo incentivo e apoio que me passou durante esta jornada. Ao Prof. Dr. Eduardo Colli, pela valiosa orientação e pela paciência por todas as vezes que precisei esclarecer algumas dúvidas. Aos professores do IME. Aos porfessores do Departamento de Matemática da UNESP S. J. do Rio Preto, pela formação e estímulo que me passaram durante a graduação.
We present some global dynamical aspects of Shimizu-Morioka equations given bẏ x = y,ẏ = x − λy −... more We present some global dynamical aspects of Shimizu-Morioka equations given bẏ x = y,ẏ = x − λy − xz,ż = −αz + x 2 , where (x, y, z) ∈ R 3 are the state variables and λ, α are real parameters. This system is a simplified model proposed for studying the dynamics of the well-known Lorenz system for large Rayleigh numbers. Using the Poincaré compactification of a polynomial vector field in R 3 , we give a complete description of the dynamics of Shimizu-Morioka equations at infinity. Then using analytical and numerical tools, we investigate for the case α = 0 the existence of infinitely many singularly degenerate heteroclinic cycles, each one consisting of an invariant set formed by a line of equilibria together with a heteroclinic orbit connecting two of these equilibria. The dynamical consequences of the existence of these cycles are also investigated. The present study is
We present a global dynamical analysis of the following quadratic differential systeṁ x=a(y−x),ẏ ... more We present a global dynamical analysis of the following quadratic differential systeṁ x=a(y−x),ẏ = dy − xz,ż =−bz + f x 2 + gx y, where (x, y, z) ∈ R 3 are the state variables and a, b, d, f, g are real parameters. This system has been proposed as a new type of chaotic system, having additional complex dynamical properties to the well-known chaotic systems defined in R 3 , alike Lorenz, Rössler, Chen and other. By using the Poincaré compactification for a polynomial vector field in R 3 , we study the dynamics of this system on the Poincaré ball, showing that it undergoes interesting types of bifurcations at infinity. We also investigate the existence of first integrals and study the dynamical behavior of the system on the invariant algebraic surfaces defined by these first integrals, showing the existence of families of homoclinic and heteroclinic orbits and centers contained on these invariant surfaces.
A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have... more A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper we prove hyperbolicity of renormalization acting on C 3 dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are C 1 manifolds.
We consider a quadratic Liénard equation with an unbounded homoclinic loop, which is a solution t... more We consider a quadratic Liénard equation with an unbounded homoclinic loop, which is a solution tending in forward and backward time to a non-hyperbolic equilibrium point located at infinity. Under small timeperiodic perturbation, this equilibrium becomes a normally hyperbolic line of singularities at infinity. We show that the perturbed system may present homoclinic bifurcations, leading to the existence of transverse intersections between the stable and unstable manifolds of such a normally hyperbolic line of singularities. The global study concerning the infinity is performed using the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in R 3 , whose boundary plays the role of the infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff-Smale Theorem, a complex dynamical behaviour of the perturbed system solutions in the finite part of the phase space. Numerical simulations are performed in order to illustrate this behaviour, which could be called ''the chaos arising from infinity'', since it depends on the global structure of the Liénard equation, including the points at infinity. Although applied to a particular case, the analysis presented provides a geometrical approach to study periodic perturbations of homoclinic (or heteroclinic) loops to infinity of any planar polynomial vector field.
Bulletin Of The Brazilian Mathematical Society, New Series, Aug 25, 2022
Given an one-dimensional Lorenz-like expanding map we prove that the condition P top (φ, ∂P, ℓ) <... more Given an one-dimensional Lorenz-like expanding map we prove that the condition P top (φ, ∂P, ℓ) < P top (φ, ℓ) (see, subsection 2.4 for definition), introduced by Buzzi and Sarig in ([1]) is satisfied for all continuous potentials φ : [0, 1] −→ R. We apply this to prove that quasi-Höldercontinuous potentials (see, subsection 2.2 for definition) have at most one equilibrium measure and we construct a family of continuous but not Hölder and neither weak Hölder continuous potentials for which we observe phase transitions. Indeed, this class includes all Hölder and weak-Hölder continuous potentials and form an open and dense subset of C([0, 1], R), with the usual C 0 topology. This give a certain generalization of the results proved in [2].
In this work we establish the relation between a geometrical hypothesis in the dynamic of interva... more In this work we establish the relation between a geometrical hypothesis in the dynamic of interval functions (with one criticam point) and the chaotic behaviour of the dynamic, defined by ergodic properties. At the some time, we show the importance of the geometric hypothesis to obtain conclusions involving the parameter, when dealing with families of mapa, implying abundance and prevalence, in the senso of Lebesgue measure, of chaotic phenomerlon outside regular hyperbolicity.
Zeitschrift für Angewandte Mathematik und Physik, Feb 28, 2017
In this paper, some ergodic aspects of non-smooth vector fields are studied. More specifically, t... more In this paper, some ergodic aspects of non-smooth vector fields are studied. More specifically, the concepts of recurrence and invariance of a measure by a flow are discussed, and two versions of the classical Poincaré Recurrence Theorem are presented. The results allow us to soften the hypothesis of the classical Poincaré Recurrence Theorem by admitting non-smooth multivalued flows. The methods used in order to prove the results involve elements from both measure theory and topology.
Bulletin of the Brazilian Mathematical Society, New Series
Given an one-dimensional Lorenz-like expanding map we prove that the condition P top (φ, ∂P, ℓ) <... more Given an one-dimensional Lorenz-like expanding map we prove that the condition P top (φ, ∂P, ℓ) < P top (φ, ℓ) (see, subsection 2.4 for definition), introduced by Buzzi and Sarig in ([1]) is satisfied for all continuous potentials φ : [0, 1] −→ R. We apply this to prove that quasi-Höldercontinuous potentials (see, subsection 2.2 for definition) have at most one equilibrium measure and we construct a family of continuous but not Hölder and neither weak Hölder continuous potentials for which we observe phase transitions. Indeed, this class includes all Hölder and weak-Hölder continuous potentials and form an open and dense subset of C([0, 1], R), with the usual C 0 topology. This give a certain generalization of the results proved in [2].
In this work we establish the relation between a geometrical hypothesis in the dynamic of interva... more In this work we establish the relation between a geometrical hypothesis in the dynamic of interval functions (with one criticam point) and the chaotic behaviour of the dynamic, defined by ergodic properties. At the some time, we show the importance of the geometric hypothesis to obtain conclusions involving the parameter, when dealing with families of mapa, implying abundance and prevalence, in the senso of Lebesgue measure, of chaotic phenomerlon outside regular hyperbolicity.
A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have... more A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2020
We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard d... more We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard differential systems \dot{x}=y, \quad \dot{y}=(ax+b)y+cx^2+dx+e, where (x, y) ∈ ℝ2 are the variables and a,b,c,d,e are real parameters.
We consider an n-dimensional piecewise smooth vector field with two zones separated by a hyperpla... more We consider an n-dimensional piecewise smooth vector field with two zones separated by a hyperplane which admits an invariant hyperplane transversal to containing a period annulus A fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in A. When n = 3w e provide normal forms for the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the given normal forms.
In this paper we consider the linear differential center (ẋ,ẏ) = (−y, x) perturbed inside the cla... more In this paper we consider the linear differential center (ẋ,ẏ) = (−y, x) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0. We provide sufficient conditions to ensure the existence of a limit cycle bifurcating from the infinity. The main tools used are the Bendixson transformation and the averaging theory.
In this paper we characterize all possible sets of periods of homeomorphisms defined on some clas... more In this paper we characterize all possible sets of periods of homeomorphisms defined on some classes of finite connected compact graphs.
Motivated by return maps near saddles for three-dimensional flows and also by return maps in the ... more Motivated by return maps near saddles for three-dimensional flows and also by return maps in the torus associated to Cherry flows, we study gap maps with derivative positive and smaller than one outside the discontinuity point. We prove that the lamination of infinitely renormalizable maps (or else maps with irrational rotation numbers) has analytic leaves in a natural subset of a Banach space of analytic maps of this kind. With maps having Hölder continuous derivative and derivative bounded away from zero, we also prove Hölder continuity of holonomies of the lamination and also of conjugacies between maps having the same combinatorics.
Page 1. Renormalization operator for affine dissipative Lorenz maps Márcio Alves∗ and Eduardo Col... more Page 1. Renormalization operator for affine dissipative Lorenz maps Márcio Alves∗ and Eduardo Colli March 14, 2008 Abstract We study properties of the renormalization operator arising in a three-dimensional family of affine dissipative Lorenz maps. ...
We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard d... more We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard differential systemṡ x = y,ẏ = (ax + b)y + cx 2 + dx + e, where (x, y) ∈ R 2 are the variables and a, b, c, d, e are real parameters.
Ao término deste trabalho, gostaria de manifestar meus sinceros agradecimentos: A minha mãe pelo ... more Ao término deste trabalho, gostaria de manifestar meus sinceros agradecimentos: A minha mãe pelo incentivo e apoio que me passou durante esta jornada. Ao Prof. Dr. Eduardo Colli, pela valiosa orientação e pela paciência por todas as vezes que precisei esclarecer algumas dúvidas. Aos professores do IME. Aos porfessores do Departamento de Matemática da UNESP S. J. do Rio Preto, pela formação e estímulo que me passaram durante a graduação.
We present some global dynamical aspects of Shimizu-Morioka equations given bẏ x = y,ẏ = x − λy −... more We present some global dynamical aspects of Shimizu-Morioka equations given bẏ x = y,ẏ = x − λy − xz,ż = −αz + x 2 , where (x, y, z) ∈ R 3 are the state variables and λ, α are real parameters. This system is a simplified model proposed for studying the dynamics of the well-known Lorenz system for large Rayleigh numbers. Using the Poincaré compactification of a polynomial vector field in R 3 , we give a complete description of the dynamics of Shimizu-Morioka equations at infinity. Then using analytical and numerical tools, we investigate for the case α = 0 the existence of infinitely many singularly degenerate heteroclinic cycles, each one consisting of an invariant set formed by a line of equilibria together with a heteroclinic orbit connecting two of these equilibria. The dynamical consequences of the existence of these cycles are also investigated. The present study is
We present a global dynamical analysis of the following quadratic differential systeṁ x=a(y−x),ẏ ... more We present a global dynamical analysis of the following quadratic differential systeṁ x=a(y−x),ẏ = dy − xz,ż =−bz + f x 2 + gx y, where (x, y, z) ∈ R 3 are the state variables and a, b, d, f, g are real parameters. This system has been proposed as a new type of chaotic system, having additional complex dynamical properties to the well-known chaotic systems defined in R 3 , alike Lorenz, Rössler, Chen and other. By using the Poincaré compactification for a polynomial vector field in R 3 , we study the dynamics of this system on the Poincaré ball, showing that it undergoes interesting types of bifurcations at infinity. We also investigate the existence of first integrals and study the dynamical behavior of the system on the invariant algebraic surfaces defined by these first integrals, showing the existence of families of homoclinic and heteroclinic orbits and centers contained on these invariant surfaces.
A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have... more A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper we prove hyperbolicity of renormalization acting on C 3 dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are C 1 manifolds.
We consider a quadratic Liénard equation with an unbounded homoclinic loop, which is a solution t... more We consider a quadratic Liénard equation with an unbounded homoclinic loop, which is a solution tending in forward and backward time to a non-hyperbolic equilibrium point located at infinity. Under small timeperiodic perturbation, this equilibrium becomes a normally hyperbolic line of singularities at infinity. We show that the perturbed system may present homoclinic bifurcations, leading to the existence of transverse intersections between the stable and unstable manifolds of such a normally hyperbolic line of singularities. The global study concerning the infinity is performed using the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in R 3 , whose boundary plays the role of the infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff-Smale Theorem, a complex dynamical behaviour of the perturbed system solutions in the finite part of the phase space. Numerical simulations are performed in order to illustrate this behaviour, which could be called ''the chaos arising from infinity'', since it depends on the global structure of the Liénard equation, including the points at infinity. Although applied to a particular case, the analysis presented provides a geometrical approach to study periodic perturbations of homoclinic (or heteroclinic) loops to infinity of any planar polynomial vector field.
Bulletin Of The Brazilian Mathematical Society, New Series, Aug 25, 2022
Given an one-dimensional Lorenz-like expanding map we prove that the condition P top (φ, ∂P, ℓ) <... more Given an one-dimensional Lorenz-like expanding map we prove that the condition P top (φ, ∂P, ℓ) < P top (φ, ℓ) (see, subsection 2.4 for definition), introduced by Buzzi and Sarig in ([1]) is satisfied for all continuous potentials φ : [0, 1] −→ R. We apply this to prove that quasi-Höldercontinuous potentials (see, subsection 2.2 for definition) have at most one equilibrium measure and we construct a family of continuous but not Hölder and neither weak Hölder continuous potentials for which we observe phase transitions. Indeed, this class includes all Hölder and weak-Hölder continuous potentials and form an open and dense subset of C([0, 1], R), with the usual C 0 topology. This give a certain generalization of the results proved in [2].
In this work we establish the relation between a geometrical hypothesis in the dynamic of interva... more In this work we establish the relation between a geometrical hypothesis in the dynamic of interval functions (with one criticam point) and the chaotic behaviour of the dynamic, defined by ergodic properties. At the some time, we show the importance of the geometric hypothesis to obtain conclusions involving the parameter, when dealing with families of mapa, implying abundance and prevalence, in the senso of Lebesgue measure, of chaotic phenomerlon outside regular hyperbolicity.
Zeitschrift für Angewandte Mathematik und Physik, Feb 28, 2017
In this paper, some ergodic aspects of non-smooth vector fields are studied. More specifically, t... more In this paper, some ergodic aspects of non-smooth vector fields are studied. More specifically, the concepts of recurrence and invariance of a measure by a flow are discussed, and two versions of the classical Poincaré Recurrence Theorem are presented. The results allow us to soften the hypothesis of the classical Poincaré Recurrence Theorem by admitting non-smooth multivalued flows. The methods used in order to prove the results involve elements from both measure theory and topology.
Bulletin of the Brazilian Mathematical Society, New Series
Given an one-dimensional Lorenz-like expanding map we prove that the condition P top (φ, ∂P, ℓ) <... more Given an one-dimensional Lorenz-like expanding map we prove that the condition P top (φ, ∂P, ℓ) < P top (φ, ℓ) (see, subsection 2.4 for definition), introduced by Buzzi and Sarig in ([1]) is satisfied for all continuous potentials φ : [0, 1] −→ R. We apply this to prove that quasi-Höldercontinuous potentials (see, subsection 2.2 for definition) have at most one equilibrium measure and we construct a family of continuous but not Hölder and neither weak Hölder continuous potentials for which we observe phase transitions. Indeed, this class includes all Hölder and weak-Hölder continuous potentials and form an open and dense subset of C([0, 1], R), with the usual C 0 topology. This give a certain generalization of the results proved in [2].
In this work we establish the relation between a geometrical hypothesis in the dynamic of interva... more In this work we establish the relation between a geometrical hypothesis in the dynamic of interval functions (with one criticam point) and the chaotic behaviour of the dynamic, defined by ergodic properties. At the some time, we show the importance of the geometric hypothesis to obtain conclusions involving the parameter, when dealing with families of mapa, implying abundance and prevalence, in the senso of Lebesgue measure, of chaotic phenomerlon outside regular hyperbolicity.
A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have... more A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2020
We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard d... more We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard differential systems \dot{x}=y, \quad \dot{y}=(ax+b)y+cx^2+dx+e, where (x, y) ∈ ℝ2 are the variables and a,b,c,d,e are real parameters.
We consider an n-dimensional piecewise smooth vector field with two zones separated by a hyperpla... more We consider an n-dimensional piecewise smooth vector field with two zones separated by a hyperplane which admits an invariant hyperplane transversal to containing a period annulus A fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in A. When n = 3w e provide normal forms for the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the given normal forms.
In this paper we consider the linear differential center (ẋ,ẏ) = (−y, x) perturbed inside the cla... more In this paper we consider the linear differential center (ẋ,ẏ) = (−y, x) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0. We provide sufficient conditions to ensure the existence of a limit cycle bifurcating from the infinity. The main tools used are the Bendixson transformation and the averaging theory.
In this paper we characterize all possible sets of periods of homeomorphisms defined on some clas... more In this paper we characterize all possible sets of periods of homeomorphisms defined on some classes of finite connected compact graphs.
Motivated by return maps near saddles for three-dimensional flows and also by return maps in the ... more Motivated by return maps near saddles for three-dimensional flows and also by return maps in the torus associated to Cherry flows, we study gap maps with derivative positive and smaller than one outside the discontinuity point. We prove that the lamination of infinitely renormalizable maps (or else maps with irrational rotation numbers) has analytic leaves in a natural subset of a Banach space of analytic maps of this kind. With maps having Hölder continuous derivative and derivative bounded away from zero, we also prove Hölder continuity of holonomies of the lamination and also of conjugacies between maps having the same combinatorics.
Page 1. Renormalization operator for affine dissipative Lorenz maps Márcio Alves∗ and Eduardo Col... more Page 1. Renormalization operator for affine dissipative Lorenz maps Márcio Alves∗ and Eduardo Colli March 14, 2008 Abstract We study properties of the renormalization operator arising in a three-dimensional family of affine dissipative Lorenz maps. ...
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