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type: Article
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aigaionid: 4191
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bibtexid: CorCorLliMoe2015
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title: Bifurcation of relative equilibria of the (1'3)-body problem
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author: Corbera, Montserrat
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author: Cors, Josep Maria
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author: Llibre, Jaume
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author: Moeckel, Richard
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journal: SIAM Journal on Mathematical Analysis
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year: 2015
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volume: 47
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number: 2
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startpage: 1377
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endpage: 1404
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doi: 10.1137/140978661
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keywords: (1 n)-body problem
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keywords: Celestial Mechanics
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keywords: relative equilibria
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abstract: We study the relative equilibria of the limit case of the planar Newtonian 4-body problem when three masses tend to zero, the so-called (1 3)-body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Six of these relative equilibria are always convex, and the others are concave. Each convex relative equilibrium of the (1 3)-body problem can be continued to a unique family of relative equilibria of the general 4-body problem when three of the masses are sufficiently small and every convex relative equilibrium for these masses belongs to one of these six families.
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file: CorCorLliMoe2015.Preprint.pdf
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type: Article
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aigaionid: 4183
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bibtexid: CanFagGar2015
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title: On a Family of Rational Perturbations of the Doubling Map
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author: Canela, Jordi
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author: Fagella, Nuria
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author: Garijo, Antoni
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journal: Journal of Difference Equations and Applications
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year: 2015
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doi: 10.1080/10236198.2015.1050387
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keywords: Blaschke products
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keywords: circle maps
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keywords: Holomorphic dynamics
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keywords: polynomial-like mappings
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file: CanFagGar2015.preprint.pdf
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type: Article
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aigaionid: 4178
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bibtexid: Gas2015b
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title: L?infinit i m?s enll?
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author: Gasull, Armengol
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journal: Materials Matem?tics
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year: 2015
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volume: 2015
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file: Gas2015b.preprint.pdf
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type: Article
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aigaionid: 4177
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bibtexid: CimGasMan2015
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title: Non-integrability of measure preserving maps via {L}ie symmetries
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author: Cima, Anna
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author: Gasull, Armengol
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author: Ma?osa, V?ctor
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journal: Journal of Differential Equations
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year: 2015
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keywords: Cohen map
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keywords: difference equations
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keywords: Integrability and non-integrability of maps
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keywords: integrable vector fields
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keywords: Isochronous center
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keywords: Lie symmetries
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keywords: measure preserving maps
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keywords: Period function
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abstract: We consider the problem of characterizing, for certain natural number m, the local C^m-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the local non-integrability of the Cohen map and of several rational maps coming from second order difference equations.
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file: CimGasMan2015.preprint.pdf
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type: Article
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aigaionid: 4176
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bibtexid: GraLli2015
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title: Divergence and {P}oincar?-{L}iapunov constants for analytic differential systems
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author: Grau, Maite
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author: Llibre, Jaume
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journal: Journal of Differential Equations
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year: 2015
|
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volume: 258
|
|
startpage: 4348
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endpage: 4367
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doi: 10.1016/j.jde.2015.01.035
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keywords: Center problem
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keywords: divergence
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keywords: Hamiltonian
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keywords: Poincar?-Liapunov constants
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abstract: We consider a planar autonomous real analytic differential system with a monodromic singular point $p$. We deal with the center problem for the singular point $p$. Our aim is to highlight some relations between the divergence of the system and the Poincar\'e-Liapunov constants of $p$ when these are defined.
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file: GraLli2015.preprint.pdf
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type: Article
|
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aigaionid: 4160
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|
bibtexid: FedPan2014
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title: The {P}icard-{F}uchs equations for complete hyperelliptic integrals of even order curves, and the actions of the generalized {N}eumann system
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author: Fedorov, Yuri
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author: Pantazi, Chara
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journal: Journal of Mathematical Physics
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year: 2014
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volume: 55
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|
startpage: 032703
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|
doi: 10.1063/1.4868965
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abstract: We consider a family of genus 2 hyperelliptic curves of even order and obtain explicitly the systems of 5 linear ordinary differential equations for periods of the corresponding Abelian integrals of first, second, and third kind, as functions of some parameters of the curves. The systems can be regarded as extensions of the well-studied Picard-Fuchs equations for periods of complete integrals of first and second kind on odd hyperelliptic curves. The periods we consider are linear combinations of the action variables of several integrable systems, in particular the generalized Neumann system with polynomial separable potentials. Thus the solutions of the extended Picard-Fuchs equations can be used to study various properties of the actions.
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file: FedPan2014.preprint.pdf
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type: Article
|
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aigaionid: 4152
|
|
bibtexid: Cau2015
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title: Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields
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author: Caubergh, Magdalena
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journal: Journal of Differential Equations
|
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year: 2015
|
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volume: 259
|
|
startpage: 989
|
|
endpage: 1013
|
|
doi: 10.1016/j.jde.2015.02.036
|
|
keywords: global phase portrait
|
|
keywords: Hilbert?s 16th Problem
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|
keywords: Limit cycle
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keywords: nilpotent center problem
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keywords: rotated vector field
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keywords: separatrix skeleton
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|
abstract: This article deals with the bifurcation of polycycles and limit cycles within the 1-parameter families of planar vector fields $X_m^k,$ defined by $\dot{x}=y^3-x^{2k 1},\dot{y}=-x my^{4k 1},$ where $m$ is a real parameter and $k\ge1$ integer. The bifurcation diagram for the separatrix skeleton of $X_m^k$ in function of $m$ is determined and the one for the global phase portraits of $(X^1_m)_{m\in\mathbb{R}}$ is completed. Furthermore for arbitrary $k\ge1$ some
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bifurcation and finiteness problems of periodic orbits are solved. Among others, the number of periodic orbits of $X_m^k$ is found to be uniformly bounded independent of $m\in\mathbb{R}$ and the Hilbert number for $(X_m^k)_{m\in\mathbb{R}},$ that thus is finite, is found to be at least one.
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file: Cau2015.preprint.pdf
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type: Article
|
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aigaionid: 4151
|
|
bibtexid: LiaLliTor2015
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|
title: Limit cycles coming from some uniform isochronous centers
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author: Liang, Haihua
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author: Llibre, Jaume
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author: Torregrosa, Joan
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|
journal: Advanced Nonlinear Studies
|
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year: 2015
|
|
keywords: averaging theory
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|
keywords: periodic solution
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|
keywords: uniform isochronous centers
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|
keywords: weak Hilbert problem
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|
abstract: This article is about the weak 16--th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed
|
|
inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers
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|
\[
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|
\dot x= -y x^2 y (x^2 y^2)^n, \dot y= x x y^2 (x^2 y^2)^n,
|
|
\]
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of degree $2n 3$ and we perturb them inside the class of all polynomial differential systems of degree $2n 3$. For $n=0,1$ we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For $n=2$ we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center.
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file: LiaLliTor2015.Preprint.pdf
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|
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type: Article
|
|
aigaionid: 4149
|
|
bibtexid: LliVal2015a
|
|
title: The co-circular central configurations of the 5 body problem
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author: Llibre, Jaume
|
|
author: Valls, Cl?udia
|
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journal: Journal of Dynamics and Differential Equations
|
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year: 2015
|
|
volume: 27
|
|
startpage: 55
|
|
endpage: 67
|
|
doi: 10.1007/s10884-015-9429-y
|
|
keywords: 5-body problem
|
|
keywords: Central configuration
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|
keywords: Co-circular central configuration
|
|
keywords: Regular n-gon
|
|
abstract: Chenciner in 2001 asked: Is the regular n?gon with equal masses the unique central configuration such that all the bodies lie on a circle, and the center of mass coincides with the center of the circle? This question has a positive answer for n = 3. Hampton in 2003 proved that also this question has a positive answer for n = 4. Here we provide a positive answer for n = 5.
|
|
file: LliVal2015a.Preprint.pdf
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|
|
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|
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type: Article
|
|
aigaionid: 4148
|
|
bibtexid: AlvLabMur2014
|
|
title: Limit cycles for a class of quintic $\mathbb {Z}_6$ equivariant systems without infinite critical points
|
|
author: ?lvarez, Maria Jes?s
|
|
author: Laboriau, Isabel S.
|
|
author: Murza, Adrian
|
|
journal: Bulletin of the Belgian Mathematical Society. Simon Stevin
|
|
year: 2014
|
|
volume: 21
|
|
startpage: 841
|
|
endpage: 857
|
|
keywords: Limit cycles
|
|
keywords: Planar autonomous ordinary differential equations
|
|
keywords: symmetric polinomial systems
|
|
abstract: We analyze the dynamics of a class of $\mathbb{Z}_6$-equivariant systems of the form $\dot{z}=pz^2\bar{z} sz^3\bar{z}^2-\bar{z}^{5},$ where $z$ is complex, the time $t$ is real, while $p$ and $s$
|
|
are complex parameters. This study is the natural continuation of a previous work (M.J. \'Alvarez, A. Gasull, R. Prohens, Proc. Am. Math. Soc. \textbf{136}, (2008), 1035--1043) on the normal form of $\mathbb{Z}_4$-equivariant systems. Our study uses the reduction of the equation to an Abel one,
|
|
and provide criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle surrounding either 1, 7 or 13 critical points, the origin being always one of these points.
|
|
file: AlvLabMur2014.Preprint.pdf
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|
|
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|
|
type: Article
|
|
aigaionid: 4147
|
|
bibtexid: GeyVil2015
|
|
title: On the wave length of smooth periodic traveling waves of the {C}amassa-{H}olm equation
|
|
author: Geyer, Anna
|
|
author: Villadelprat, Jordi
|
|
journal: Journal of Differential Equations
|
|
year: 2015
|
|
volume: 259
|
|
startpage: 2317
|
|
endpage: 2332
|
|
doi: http://dx.doi.org/10.1016/j.jde.2015.03.027
|
|
abstract: This paper is concerned with the wave length $\lambda$ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height $a$ (or ``peak-to-peak amplitude''). Our main result establishes monotonicity properties of the map $a\longmapsto \lambda(a)$, i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated to the equation, which distinguish between the two possible qualitative behaviours of $\lambda(a)$, namely monotonicity and unimodality. The key point is to relate $\lambda(a)$ to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.
|
|
file: GeyVil2015.Preprint.pdf
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|
|
|
|
|
type: Article
|
|
aigaionid: 4144
|
|
bibtexid: Gas2015
|
|
title: B?tes i barrils
|
|
author: Gasull, Armengol
|
|
journal: Nou Biaix
|
|
year: 2015
|
|
file: Gas2015.Preprint.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4143
|
|
bibtexid: LliPes2015
|
|
title: The {H}opf bifurcation in the {S}himizu-{M}orioka system
|
|
author: Llibre, Jaume
|
|
author: Pessoa, Claudio
|
|
journal: Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems
|
|
year: 2015
|
|
volume: 79
|
|
startpage: 2197
|
|
endpage: 2205
|
|
doi: 10.1007/s11071-014-1805-3
|
|
keywords: bifurcation diagram
|
|
keywords: Hopf bifurcation
|
|
keywords: Limit cycles
|
|
file: LliPes2015.Preprint.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4135
|
|
bibtexid: ArtLliSchVul2015
|
|
title: Global configurations of singularities for quadratic differential systems with exactly three finite singularities of total multiplicity four
|
|
author: Art?s, Joan Carles
|
|
author: Llibre, Jaume
|
|
author: Schlomiuk, Dana
|
|
author: Vulpe, Nicolae
|
|
journal: The Rocky Mountain Journal of Mathematics
|
|
year: 2015
|
|
volume: 45
|
|
number: 1
|
|
startpage: 29
|
|
endpage: 113
|
|
keywords: affine invariant polynomials
|
|
keywords: configuration of singularities
|
|
keywords: geometric equivalence relation
|
|
keywords: infinite and finite singularities
|
|
keywords: Poincar? compactification
|
|
keywords: Quadratic vector fields
|
|
abstract: In the topological classification of phase portraits no distinctions are made between a focus and a node and neither are they made between a strong and a weak focus or between foci of different orders. These distinction are however important in the production of limit cycles close to the foci in perturbations of the systems. The distinction between the one direction node and the two directions node, which plays a role in understanding the behavior of solution curves around the singularities at infinity, is also missing in the
|
|
topological classification.
|
|
|
|
In this work we introduce the notion of \textit{geometric equivalence relation of singularities} which incorporates these important purely algebraic features. The \textit{geometric} equivalence relation is finer than the \textit{topological} one and also finer than the \textit{qualitative equivalence relation} introduced in \cite{J_L}. We also list all possibilities we have for singularities finite and infinite taking into consideration these finer distinctions and introduce notations for each one of them. Our long term goal is to use this finer equivalence relation to classify the quadratic family according to their different \textit{geometric configurations of singularities}, finite and infinite.
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|
|
|
In this work we accomplish a first step of this larger project. We give a complete global classification, using the \textit{geometric equivalence} relation, of the whole quadratic class according to the configuration of singularities at infinity of the systems. Our classification theorem is stated in terms of invariant polynomials and hence it can be applied to any family of quadratic systems with respect to any particular normal form. The theorem we give also contains the bifurcation diagram, done in the 12-parameter space, of
|
|
the \textit{geometric configurations} of singularities at infinity, and this bifurcation set is algebraic in the parameter space. To determine the bifurcation diagram of configurations of singularities at infinity for any family of quadratic systems, given in any normal form, becomes thus a simple task using computer algebra
|
|
calculations.
|
|
file: ArtLliSchVul2015.preprint.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4130
|
|
bibtexid: EuzLli2015
|
|
title: On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line
|
|
author: D. Euz?bio, Rodrigo
|
|
author: Llibre, Jaume
|
|
journal: Journal of Mathematical Analysis and Applications
|
|
year: 2015
|
|
volume: 424
|
|
startpage: 475
|
|
endpage: 486
|
|
doi: 10.1016/j.jmaa.2014.10.077
|
|
keywords: Limit cycle
|
|
keywords: nonsmooth differential system
|
|
keywords: piecewise linear differential system
|
|
file: EuzLli2015.preprint.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4127
|
|
bibtexid: LliLopMor2015
|
|
title: Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2
|
|
author: Llibre, Jaume
|
|
author: Lopes, Bruno D.
|
|
author: de Moraes, Jaime R.
|
|
journal: Applied Mathematics and Computation
|
|
year: 2015
|
|
volume: 250
|
|
startpage: 887
|
|
endpage: 907
|
|
doi: 10.1016/j.amc.2014.11.029
|
|
keywords: Averaging method
|
|
keywords: Isochronous center
|
|
keywords: Limit cycles
|
|
keywords: periodic orbits
|
|
keywords: Polynomial vector fields
|
|
file: LliLopMor2015.Preprint.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4126
|
|
bibtexid: ArtRezOli2014a
|
|
title: The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (C)
|
|
author: Art?s, Joan Carles
|
|
author: Rezende, Alex Carlucci
|
|
author: Oliveira, Regilene D. S.
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
year: 2015
|
|
keywords: algebraic invariants
|
|
keywords: bifurcation diagram
|
|
keywords: finite saddle-node
|
|
keywords: infinite saddle-node
|
|
keywords: Phase portrait
|
|
keywords: Quadratic differential systems
|
|
file: ArtRezOli2014a.preprint.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4125
|
|
bibtexid: ArtLliRezSchVul2014
|
|
title: Global configurations of singularities for quadratic differential systems with exactly two finite singularities of total multiplicity four
|
|
author: Art?s, Joan Carles
|
|
author: Llibre, Jaume
|
|
author: Rezende, Alex Carlucci
|
|
author: Schlomiuk, Dana
|
|
author: Vulpe, Nicolae
|
|
journal: Electronic Journal of Qualitative Theory of Differential Equations
|
|
year: 2014
|
|
volume: 60
|
|
startpage: 1
|
|
endpage: 43
|
|
keywords: affine invariant polynomials
|
|
keywords: configuration of singularities
|
|
keywords: geometric equivalence relation
|
|
keywords: infinite and finite singularities
|
|
keywords: Poincar? compactification
|
|
keywords: Quadratic vector fields
|
|
file: ArtLliRezSchVul2014.preprint.pdf
|