|
<entry>
|
|
type: Phdthesis
|
|
aigaionid: 3474
|
|
bibtexid: Per2012
|
|
title: On the number of limit cycles for some families of planar differential equations
|
|
author: Pérez-González, Set
|
|
year: 2012
|
|
school: Universitat Autònoma de Barcelona
|
|
address: Bellaterra (Barcelona)
|
|
file: Per2012.pdf-5e3f269f95266208c9e1351a7404c1df.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3412
|
|
bibtexid: BuzLli2012a
|
|
title: On the periodic solutions of the static, spherically symmetric {E}instein-{Y}ang-{M}ills equations
|
|
author: Buzzi, Claudio Aguinaldo
|
|
author: Llibre, Jaume
|
|
journal: Journal of Mathematical Physics
|
|
year: 2012
|
|
volume: 53
|
|
startpage: 122703
|
|
file: BuzLli2012a.pdf-58b7545892b6a81c1f37aaa8183d0d80.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Phdthesis
|
|
aigaionid: 3607
|
|
bibtexid: Bol2013
|
|
title: Generalizations of the {D}arboux integrability theory for polynomial vector fields
|
|
author: Bolaños Rivera, Yudi Marcela
|
|
year: 2013
|
|
school: Universitat Autònoma de Barcelona
|
|
address: Bellaterra
|
|
note: In Spanish
|
|
file: Bol2013.pdf-83db7f3ff7ccb7d46bccdb469778b625.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3668
|
|
bibtexid: ArtLliSchVul2013b
|
|
title: Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities
|
|
author: Artés, Joan Carles
|
|
author: Llibre, Jaume
|
|
author: Schlomiuk, Dana
|
|
author: Vulpe, Nicolae
|
|
journal: Journal of Fixed Point Theory and Applications
|
|
year: 2013
|
|
volume: 14
|
|
number: 2
|
|
startpage: 555
|
|
endpage: 618
|
|
doi: 10.1007/s11784-014-0175-2
|
|
keywords: geometric equivalence relation
|
|
keywords: Quadratic vector fields
|
|
keywords: Singularities
|
|
abstract: In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. This relation is finer than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci (or saddles) of different orders. Such distinctions are, however, important in the production of limit cycles close to the foci (or loops) in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows us to incorporate all these important geometric features which can be expressed in purely algebraic terms. The geomet
|
|
ric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in a work published in 2013 when the classification was done for systems with total multiplicity m f of finite singularities less than or equal to one. That work was continued in an article which is due to appear in 2014 where the geometric classification of configurations of singularities was done for the case m f = 2. In this article we go one step further and obtain the geometric classification of singularities, finite and infinite, for the subclass mentioned above. We
|
|
obtain 147 geometrically distinct configurations of singularities for this family. We give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants, a fact which gives us an algorithm for determining the geometric configuration of singularities for any quadratic system in this
|
|
particular class.
|
|
file: ArtLliSchVul2013.pdf-d07a39d224471f46d61731058071cac7.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Phdthesis
|
|
aigaionid: 4411
|
|
bibtexid: Gey2013
|
|
title: On some aspects of nonlinear water wave theory
|
|
author: Geyer, Anna
|
|
year: 2013
|
|
school: University of Vienna
|
|
address: Vienna
|
|
file: Gey2013.pdf-9799ad0bbfc602dfef4cc084dbec7862.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 2337
|
|
bibtexid: CaoLliZha2014
|
|
title: Darboux integrability and Algebraic limit cycles for a class of polynomial differential Systems
|
|
author: Cao, Jinlong
|
|
author: Llibre, Jaume
|
|
author: Zhang, Xiang
|
|
journal: Science China Mathematics
|
|
year: 2014
|
|
volume: 57
|
|
number: 4
|
|
startpage: 775
|
|
endpage: 794
|
|
doi: 10.1007/s11425-014-4772-8
|
|
keywords: Algebraic limit cycles
|
|
keywords: Darboux first integral
|
|
file: CaoLliZha2008.pdf-5d4cbd46028da8ae.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Phdthesis
|
|
aigaionid: 4109
|
|
bibtexid: Col2014
|
|
title: Hamiltonian linear type centers and nilpotent centers of linear plus cubic polynomial vector fields
|
|
author: Colak, Ilker
|
|
year: 2014
|
|
school: Universitat Autònoma de Barcelona
|
|
address: Bellaterra (Barcelona)
|
|
keywords: Center
|
|
keywords: cubic Hamiltonian
|
|
file: Col2014.pdf-6fca9c12c74bd08fa038bcc4a008a3d2.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Mastersthesis
|
|
aigaionid: 4347
|
|
bibtexid: Gou2014
|
|
title: Classification of centers and study of limit cycles for piecewise linear systems in two zones on the plane
|
|
author: Gouveia, Luiz Fernando da Silva
|
|
year: 2014
|
|
school: Universidade Estadual Paulista "Júlio de Mesquita Filho"
|
|
address: Sao Jose do Rio Preto
|
|
note: In Portuguese
|
|
file: Gou2014.pdf-20a8304a0962eb45c2bb6ccaebb5dbc8.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Phdthesis
|
|
aigaionid: 4412
|
|
bibtexid: Zaf2014
|
|
title: Dynamical Classification of some Birational Maps of ${C}^2$
|
|
author: Zafar, Sundus
|
|
year: 2014
|
|
school: Univertiat Autònoma de Barcelona
|
|
address: Bellaterra
|
|
file: Zaf2014.pdf-cb733cb1342a34e5b4a385a33d986684.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Phdthesis
|
|
aigaionid: 4413
|
|
bibtexid: Gar2014b
|
|
title: A qualitative and quantitative study of some planar differential equations
|
|
author: García Saldaña, Johanna Denise
|
|
year: 2014
|
|
school: Universitat Autònoma de Barcelona
|
|
address: Bellaterra
|
|
file: Gar2014.pdf-0f845db272411cc3404ddf602746dba7.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4142
|
|
bibtexid: LliLon2015
|
|
title: Periodic solutions for the generalized anisotropic {L}ennard-{J}ones {H}amiltonian
|
|
author: Llibre, Jaume
|
|
author: Long, Yiming
|
|
journal: Qualitative Theory of Dynamical Systems
|
|
year: 2015
|
|
volume: 14
|
|
startpage: 291
|
|
endpage: 311
|
|
doi: 10.1007/s12346-015-0167-7
|
|
keywords: anisotropic Lennard-Jones potential
|
|
keywords: circular periodic solutions
|
|
keywords: Lennard-Jones potential
|
|
file: LliLon2015.preprint.pdf-80a5fbedf6b8204c9b8f7e2b3630c916.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Incollection
|
|
aigaionid: 4333
|
|
bibtexid: BarCopr2015
|
|
title: Convex Central Configurations of Two Twisted n-gons
|
|
author: Barrabes, Esther
|
|
author: Cors, Josep Maria
|
|
booktitle: Extended Abstracts Spring 2014 (Hamiltonian Systems and Celestial Mechanics, Virus Dynamics and Evolution)
|
|
series: Trends in Mathematics. Research Perspectives CRM Barcelona
|
|
year: 2015
|
|
volume: 4
|
|
startpage: 17
|
|
endpage: 21
|
|
publisher: Birkhäuser
|
|
doi: 10.1007/978-3-319-22129-8_3
|
|
file: BarCor2015.Preprint.pdf-3e0b94a1be3fd5530a1593525b6f6e3d.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3438
|
|
bibtexid: Lli2015c
|
|
title: On the 16-{H}ilbert Problem
|
|
author: Llibre, Jaume
|
|
journal: La Gaceta de la Real Sociedad Matemática Española
|
|
year: 2015
|
|
volume: 18
|
|
number: 3
|
|
startpage: 543
|
|
endpage: 554
|
|
note: In Spanish
|
|
abstract: Presentamos un breve resumen de algunos resultados recientes sobre la segunda parte del problema 16 de Hilbert, poniendo un especial énfasis en los ciclos límite algebraicos.
|
|
file: Lli2012.pdf-2879fca33a006d0e9d7ae091e9574c03.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3539
|
|
bibtexid: LliVid2015
|
|
title: Periodic solutions of a periodic {F}itz{H}ugh-{N}agumo differential system
|
|
author: Llibre, Jaume
|
|
author: Vidal, Claudio
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
year: 2015
|
|
volume: 25
|
|
number: 13
|
|
startpage: 1550180 (6 pages)
|
|
doi: 10.1142/S0218127415501801
|
|
keywords: Averaging theory
|
|
keywords: FitzHugh-Nagumo system
|
|
keywords: periodic orbit
|
|
abstract: Recently some interest has appeared for the periodic FitzHugh–Nagumo differential systems. Here, we provide sufficient conditions for the existence of periodic solutions in such differential systems.
|
|
file: LliVid2013.pdf-042bf8e1e52b3438535fd924b5f3300a.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3550
|
|
bibtexid: BolLliVal2015
|
|
title: Liouvillian first integrals for quadratic systems with an integrable saddle
|
|
author: Bolaños Rivera, Yudi Marcela
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: The Rocky Mountain Journal of Mathematics
|
|
year: 2015
|
|
volume: 45
|
|
number: 6
|
|
startpage: 1765
|
|
endpage: 1779
|
|
doi: 10.1216/RMJ-2015-45-6-1765
|
|
keywords: integrable saddle
|
|
keywords: Integrating factor
|
|
keywords: Inverse integrating factor
|
|
keywords: Liouvillian first integral
|
|
keywords: quadratic systems
|
|
abstract: We provide explicit expressions for the Liouvillian first integrals of the quadratic polynomial differential systems having an integrable saddle.
|
|
file: BolLliVal2013b.pdf-ed17a96e763f27b868a1f65dbaaa9d1a.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3687
|
|
bibtexid: LliVal2015l
|
|
title: Analytic integrability of a class of planar polynomial differential systems
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Discrete and Continuous Dynamical Systems. Series B
|
|
year: 2015
|
|
volume: 20
|
|
number: 8
|
|
startpage: 2657
|
|
endpage: 2661
|
|
doi: 10.3934/dcdsb.2015.20.2657
|
|
file: LliVal2014b.pdf-8c5f12362cd82cace24995a949f057e4.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3718
|
|
bibtexid: GuiLliVer2016
|
|
title: Periodic orbits of a perturbed 3–dimensional isotropic oscillator with axial symmetry
|
|
author: Guirao, Juan Luis Garcia
|
|
author: Llibre, Jaume
|
|
author: Vera, Juan A.
|
|
journal: Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems
|
|
year: 2015
|
|
volume: 83
|
|
startpage: 839
|
|
endpage: 848
|
|
doi: 10.1007/s11071-015-2371-z
|
|
abstract: We study the periodic orbits of a generalized Yang–Mills Hamiltonian H depending on a parameter β. Playing with the parameter β we are considering extensions of the Contopoulos and of the Yang–Mills Hamiltonians in a 3-dimensional space. This Hamiltonian consists of a 3-dimensional isotropic harmonic oscillator plus a homogeneous potential of fourth degree having an axial symmetry, which implies that the third component N of the angular momentum is constant. We prove that in each invariant space H = h > 0 the Hamiltonian system has at least four periodic solutions if either β < 0, or β = 5 sqrt(13); and at least 12 periodic solutions if β > 6 and β != 5 sqrt(13). We also study the linear stability or instability of these periodic solutions.
|
|
file: GuiLliVer2014.preprint.pdf-3483bc0923d8387d6cee827c29d6c294.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3749
|
|
bibtexid: LliNovTei2015d
|
|
title: On the periodic solutions of perturbed 4D non-resonant systems
|
|
author: Llibre, Jaume
|
|
author: Novaes, Douglas D.
|
|
author: Teixeira, Marco Antonio
|
|
journal: Sao Paulo Journal of Mathematics
|
|
year: 2015
|
|
volume: 9
|
|
startpage: 229
|
|
endpage: 250
|
|
doi: 10.1007/s40863-015-0017-1
|
|
keywords: Averaging theory
|
|
keywords: Double pendulum
|
|
keywords: Non-resonant systems
|
|
keywords: Non-smooth dynamical systems
|
|
keywords: periodic solution
|
|
abstract: We provide sufficient conditions for the existence of periodic solutions of a 4D non-resonant system perturbed by smooth or non-smooth functions. We apply these results to study the small amplitude periodic solutions of the non-linear planar double pendulum perturbed by smooth or non-smooth function.
|
|
file: LliNovTei2014a.Preprint.pdf-6cedd7faadbbbd97275239a551194b73.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3771
|
|
bibtexid: BerFagRem2015
|
|
title: Hyperbolic entire functions with bounded {F}atou components
|
|
author: Bergweiler, Walter
|
|
author: Fagella, Nuria
|
|
author: Rempe, Lasse
|
|
journal: Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society
|
|
year: 2015
|
|
volume: 90
|
|
number: 4
|
|
startpage: 799
|
|
endpage: 829
|
|
doi: 10.4171/CMH/371
|
|
keywords: Axiom A
|
|
keywords: Bounded Fatou component
|
|
keywords: Eremenko-Lyubich class
|
|
keywords: Fatou set
|
|
keywords: Hyperbolicity
|
|
keywords: Jordan curve
|
|
keywords: Julia set
|
|
keywords: Laguerre-Pólya class
|
|
keywords: Local connectivity
|
|
keywords: Quasicircle
|
|
keywords: Quasidisc
|
|
keywords: Transcendental entire function
|
|
abstract: We show that an invariant Fatou component of a hyperbolic transcendental entire function is a Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou components and local connectivity of Julia sets for hyperbolic entire functions, and give examples that demonstrate that our results are optimal. A particularly strong dichotomy is obtained in the case of a function with precisely two critical values.
|
|
file: BerFagRem2014.preprint.pdf-b06df4e5df71eb1b445c49522463f97a.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3774
|
|
bibtexid: GarGod2015
|
|
title: On {M}c{M}ullen-like mappings
|
|
author: Garijo, Antoni
|
|
author: Godillon, Sebastién
|
|
journal: Journal of Fractal Geometry
|
|
year: 2015
|
|
volume: 2
|
|
startpage: 249
|
|
endpage: 279
|
|
doi: 10.4171/JFG/21
|
|
keywords: complex dynamics
|
|
keywords: Julia sets
|
|
keywords: McMullen family
|
|
keywords: rational maps
|
|
abstract: We introduce a generalization of the McMullen family $f_\lambda(z) = z^n \lambda/zd^$. In 1988 C. McMullen showed that the Julia set of $f_\lambda$ is a Cantor set of circles if and only if $1/n 1/d < 1$ and the simple critical values of $f_\lambda$ belong to the trap door. We generalize this behavior and we define a McMullen-like mapping as a rational map f associated to a hyperbolic postcritically finite polynomial $P$ and a pole data $\mathcal D$ where we encode, basically, the location of every pole of f and the local degree at each pole. In the McMullen family the polynomial $P$ is $z\mapsto z^n$ and the pole data $\mathcal D$ is the pole located at the origin that maps to infinity with local degree $d$. As in the McMullen family $f_\lambda$, we can characterize a McMullen-like mapping using an arithmetic condition depending only on the polynomial $P$ and the pole data $\mathcal D$. We prove that the arithmetic condition is necessary using the theory of Thurston's obstructions, and sufficient by quasiconformal surgery.
|
|
file: GarGod2014.preprint.pdf-9318b10fc87acdb693853a6ba840077a.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4112
|
|
bibtexid: Gas2015h
|
|
title: Fórmules i filatèlia
|
|
author: Gasull, Armengol
|
|
journal: Notícies de la Societat Catalana de Matemàtiques
|
|
year: 2015
|
|
volume: 37
|
|
startpage: 49
|
|
endpage: 54
|
|
file: Gas2014.preprint.pdf-8fdb34f4c9b4d26b3d8983e44a4d6507.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4126
|
|
bibtexid: ArtRezOli2015
|
|
title: The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (C)
|
|
author: Artés, 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
|
|
volume: 25
|
|
number: 3
|
|
doi: 10.1142/S0218127415300098
|
|
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-7cda85e6ca7167c8f30eace42afaec80.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4134
|
|
bibtexid: LliYu2015
|
|
title: Global phase portraits of quadratic systems with an ellipse and a straight line as invariant algebraic curves
|
|
author: Llibre, Jaume
|
|
author: Yu, Jiang
|
|
journal: Electronic Journal of Differential Equations
|
|
year: 2015
|
|
volume: 314
|
|
startpage: 1
|
|
endpage: 14
|
|
keywords: First integral
|
|
keywords: global phase portraits
|
|
keywords: invariant ellipse
|
|
keywords: invariant straight line
|
|
keywords: quadratic system
|
|
abstract: In this article we study a class of integrable quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having an ellipse and a straight line as invariant algebraic curves. We show that this class is integrable and we provide all the different topological phase portraits that this class exhibits in the Poincaré disc.
|
|
file: LliYu2014.preprint.pdf-3d8b4a35293f5fa2e6461b4f0dc6c9a7.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4144
|
|
bibtexid: Gas2015
|
|
title: Bótes i barrils
|
|
author: Gasull, Armengol
|
|
journal: Nou Biaix
|
|
year: 2015
|
|
volume: 36
|
|
startpage: 8
|
|
endpage: 28
|
|
abstract: In this paper we consider the validity of several practical formulas used to calculate the volume of wine or cider barrels. In our study we will find, for instance, Simpson’s formula for calculating definite integrals; a book written by Kepler on the subject; and the method of least squares established by Gauss and Legendre, for calculating the best solution for overdetermined and incompatible systems.
|
|
file: Gas2015.Preprint.pdf-4e84eb2ee5779ab4b29251abb6ad6bb2.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4233
|
|
bibtexid: GinLli2015
|
|
title: Canards existence in {F}itz{H}ugh-{N}agumo and {H}odgkin-{H}uxley neuronal models
|
|
author: Ginoux, Jean-Marc
|
|
author: Llibre, Jaume
|
|
journal: Mathematical Problems in Engineering
|
|
year: 2015
|
|
volume: 2015
|
|
number: 342010
|
|
startpage: 17pp.
|
|
doi: 10.1155/2015/342010
|
|
keywords: canard solutions
|
|
keywords: Geometric singular perturbation theory
|
|
keywords: singularly perturbed dynamical systems
|
|
abstract: In a previous paper we have proposed a new method for proving the existence of "canard solutions" for three and four-dimensional singularly perturbed systems with only one fast variable. The aim of this work is to extend this method to the case of four-dimensional singularly perturbed systems with two slow and two fast variables. This method enables to state a unique generic condition for the existence of "canard solutions" for such four-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. This unique generic condition is perfectly identical to that provided in previous works. Applications of this method to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model enables to show the existence of "canard solutions" in such systems.
|
|
file: GinLli2015.Preprint.pdf-4f4cccd998fdb402e9d2645d72b5f32b.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4338
|
|
bibtexid: LemLli2015
|
|
title: Periodic orbits for a generalized {F}riedmann-{R}obertson-{W}alker {H}amiltonian system in dimension 6
|
|
author: Lembarki, Fatima E.
|
|
author: Llibre, Jaume
|
|
journal: Discrete and Continuous Dynamical Systems. Series S
|
|
year: 2015
|
|
volume: 8
|
|
number: 6
|
|
startpage: 1165
|
|
endpage: 1211
|
|
doi: 10.3934/dcdss.2015.8.1165
|
|
keywords: Averaging theory
|
|
keywords: fam- ily of periodic orbits
|
|
keywords: Friedmann-Robertson-Walker
|
|
keywords: periodic orbits
|
|
keywords: periodic orbits parameterized by the energy
|
|
abstract: A generalized Friedmann-Robertson-Walker Hamiltonian system is studied in dimension 6. The averaging theory is the tool used to provide sufficient conditions on the six parameters of the system which guarantee the existence of continuous families of period orbits parameterized by the energy.
|
|
file: LemLli2015.Preprint.pdf-25aa0e4d12d874ec6da57a66e73869bc.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Inbook
|
|
aigaionid: 4346
|
|
bibtexid: Lli2015d
|
|
title: The Averaging Theory for Computing Periodic Orbits
|
|
author: Llibre, Jaume
|
|
booktitle: Central Configurations, Periodic Orbits, and Hamiltonian Systems
|
|
series: Advanced Courses in Mathematics - CRM Barcelona
|
|
year: 2015
|
|
publisher: Birkhäuser Basel
|
|
address: CRM Barcelona
|
|
isbn: 978-3-0348-0932-0
|
|
doi: 10.1007/978-3-0348-0933-7
|
|
file: Lli2015d.Preprint.pdf-0f9b7fcb859c6aecd036b8e08ebeb950.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4348
|
|
bibtexid: LliNovTei2015c
|
|
title: Periodic solutions of Lienard differential equations via averaging theory of order two
|
|
author: Llibre, Jaume
|
|
author: Novaes, Douglas D.
|
|
author: Teixeira, Marco Antonio
|
|
journal: Anais da Academia Brasileira de Ciencias
|
|
year: 2015
|
|
volume: 87
|
|
number: 4
|
|
startpage: 1905
|
|
endpage: 1913
|
|
doi: 10.1590/0001-3765201520140129
|
|
keywords: Averaging theory
|
|
keywords: bifurcation theory
|
|
keywords: Lienard differential equation
|
|
keywords: periodic solution
|
|
file: LliNovTei2015c.Preprint.pdf-4ce18f3d5a5db8fb5edde490013ed896.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Phdthesis
|
|
aigaionid: 4414
|
|
bibtexid: Can2015
|
|
title: On a Family of Degree 4 Blaschke Products
|
|
author: Canela, Jordi
|
|
year: 2015
|
|
school: Universitat de Barcelona
|
|
address: Barcelona
|
|
file: Can2015.pdf-bb908c52f17193168948c327c1d51bee.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3441
|
|
bibtexid: PerTorTor2016
|
|
title: Existence and uniqueness of limit cycles for generalized $\varphi$-{L}aplacian {L}iénard equations
|
|
author: Pérez-González, Set
|
|
author: Torregrosa, Joan
|
|
author: Torres, Pedro J.
|
|
journal: Journal of Mathematical Analysis and Applications
|
|
year: 2016
|
|
volume: 439
|
|
startpage: 745
|
|
endpage: 765
|
|
doi: http://dx.doi.org/10.1016/j.jmaa.2016.03.004
|
|
keywords: Existence and Uniqueness
|
|
keywords: Generalized Liénard equations.
|
|
keywords: limit cycles
|
|
keywords: periodic orbits
|
|
keywords: φ-Laplacian Liénard equations
|
|
file: PerTorTor2016.Preprint.pdf-8c1777e2277ba5d95b67704a4207356d.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3541
|
|
bibtexid: LliMak2016a
|
|
title: Zero-{H}opf periodic orbit of a non-autonomous quadratic differential system obtained from third-equations
|
|
author: Llibre, Jaume
|
|
author: Makhlouf, Ammar
|
|
journal: Chaos, Solitons and Fractals
|
|
year: 2016
|
|
volume: 89
|
|
startpage: 228
|
|
endpage: 231
|
|
doi: 10.1016/j.chaos.2015.11.013
|
|
keywords: Averaging theory
|
|
keywords: Michelson system
|
|
keywords: periodic solution
|
|
keywords: Triple-zero bifurcation
|
|
keywords: zero-Hopf bifurcation
|
|
abstract: We provide sufficient conditions for the existence of two periodic solutions bifurcating from a zero–Hopf equilibrium for the differential system
|
|
x ̇ =y, y ̇ =z, z ̇ =a by cz−x^2/2,
|
|
where a, b and c are real arbitrary parameters. The regular perturbation of this differential system provides the normal form of the so–called triple–zero bifurcation.
|
|
file: LliMak2013a.pdf-376659084510b753d1260f55afeb41d4.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3554
|
|
bibtexid: LliVal2016d
|
|
title: On the polynomial integrability of a system motivated by the {R}iemann ellipsoid problem
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: ESAIM. Control, Optimisation and Calculus of Variations
|
|
year: 2016
|
|
volume: 22
|
|
startpage: 872
|
|
endpage: 882
|
|
doi: 10.1051/cocv/2015035
|
|
keywords: complete integrability
|
|
keywords: Euler–Poinsot systems
|
|
keywords: homogeneous differential systems
|
|
keywords: polynomial first integral
|
|
keywords: Riemann ellipsoid problem
|
|
abstract: We consider differential systems obtained by coupling two Euler–Poinsot systems. The motivation to consider such systems can be traced back to the Riemann ellipsoid problem. We provide new cases for which these systems are completely integrable. We also prove that these systems either are completely integrable or have at most four functionally independent analytic first integrals.
|
|
file: LliVal2013h.pdf-ff4df2ae4997678daeea1d024b8e2aaf.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3723
|
|
bibtexid: Reb2016
|
|
title: Poincaré–{P}ontryagin–{M}elnikov functions for a class of perturbed planar {H}amiltonian equations
|
|
author: Rebollo-Perdomo, Salomón
|
|
journal: Qualitative Theory of Dynamical Systems
|
|
year: 2016
|
|
doi: 10.1007/s12346-015-0185-5
|
|
abstract: In this paper we consider polynomial perturbations of a family of polynomial Hamiltonian equations whose associated Hamiltonian is not transversal to infinity, and its complexification is not a Morse polynomial. We look for an algorithm to compute the first non-vanishing Poincaré–Pontryagin–Melnikov function of the displacement function associated with the perturbed equation. We show that the algorithm of the case when the Hamiltonian is transversal to infinity and its complexification is a Morse polynomial can be extended to our family of perturbed equations. We apply the result to study the maximum number of zeros of the first non-vanishing Poincaré–Pontryagin–Melnikov function associated with some perturbed Hamiltonian equations.
|
|
file: Reb-Per2014.pdf-434c014d9bb2d4bc153f6a431871c8c6.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3763
|
|
bibtexid: LliValZha2016
|
|
title: Liouvillian integrability versus {D}arboux polynomials
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
author: Zhang, Xiang
|
|
journal: Qualitative Theory of Dynamical Systems
|
|
year: 2016
|
|
volume: 15
|
|
startpage: 503
|
|
endpage: 515
|
|
doi: 10.1007/s12346-016-0212-1
|
|
keywords: Darboux Jacobian multiplier
|
|
keywords: Darboux polynomial
|
|
keywords: Liouvillian integrability
|
|
keywords: Polynomial differential systems
|
|
abstract: In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Liénard polynomial differential system $x ̇ = y, y ̇ = − f (x)y − g(x)$ with $deg f > deg g$ is not Liouvillian integrable.
|
|
file: LliValZha2014b.preprint.pdf-48ffa3ab292f530e15c5ede2c6cb9223.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 3776
|
|
bibtexid: LliTeiZel2016
|
|
title: Birth of limit cycles for a classe of continuous and discontinuous differential systems in (d 2)-dimension
|
|
author: Llibre, Jaume
|
|
author: Teixeira, Marco Antonio
|
|
author: Zeli, Iris O.
|
|
journal: Dynamical Systems. An International Journal
|
|
year: 2016
|
|
volume: 31
|
|
number: 3
|
|
startpage: 237
|
|
endpage: 250
|
|
doi: 10.1080/14689367.2015.1102868
|
|
keywords: Averaging theory
|
|
keywords: discontinuous polynomial differential system
|
|
keywords: limit cycle
|
|
keywords: periodic orbit
|
|
file: LliTeiZel2014.preprint.pdf-779911ace565ca58e8d5e8eca0e4f86c.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4015
|
|
bibtexid: LliRamRamSad2016
|
|
title: The 16th {H}ilbert problem restricted to circular algebraic limit cycles
|
|
author: Llibre, Jaume
|
|
author: Ramírez, Rafael Orlando
|
|
author: Ramírez, Valentín
|
|
author: Sadovskaia, Natalia
|
|
journal: Journal of Differential Equations
|
|
year: 2016
|
|
volume: 260
|
|
startpage: 5726
|
|
endpage: 5760
|
|
doi: 10.1016/j.jde.2015.12.019 0022-0396
|
|
keywords: Darboux integrability
|
|
keywords: invariant algebraic circles
|
|
keywords: Planar polynomial differential system
|
|
keywords: Polynomial vector fields
|
|
abstract: We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S - 1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S - 1 algebraic limit cycles given by circles, and this number is reached.
|
|
file: LliRamRamSad2014.preprint.pdf-eda8a23e48b5a8d554587ac4fce2cfef.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4070
|
|
bibtexid: FerVal2016
|
|
title: On the {D}arboux integrability of a cubic {C}{R}{N}{T} model in $\mathbb {R}^5$
|
|
author: Ferragut, Antoni
|
|
author: Valls, Clàudia
|
|
journal: Chaos, Solitons and Fractals
|
|
year: 2016
|
|
volume: 82
|
|
startpage: 131
|
|
endpage: 138
|
|
doi: 10.1016/j.chaos.2015.11.011
|
|
keywords: chemical reaction network
|
|
keywords: Darboux integrability
|
|
keywords: Darboux polynomial
|
|
keywords: Exponential factor
|
|
file: FerVal2014b.preprint.pdf-d4e85b08ae6af0125a221da8aa0d06a5.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4102
|
|
bibtexid: LliLopMor2016
|
|
title: Limit cycles bifurcating from the periodic annulus of the weight-homogeneous polynomial centers of weight-degree 2
|
|
author: Llibre, Jaume
|
|
author: Lopes, Bruno D.
|
|
author: de Moraes, Jaime R.
|
|
journal: Applied Mathematics and Computation
|
|
year: 2016
|
|
volume: 274
|
|
startpage: 47
|
|
endpage: 54
|
|
doi: 10.1016/j.amc.2015.10.079
|
|
keywords: Averaging theory
|
|
keywords: limit cycle
|
|
keywords: Polinomial vector field
|
|
keywords: weight-homogeneous differential system
|
|
abstract: We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic polynomial differential systems. The family considered is the unique family of weight-homogeneous polynomial differential systems of weight-degree 2 with a center. The computations has been done with the help of the algebraic manipulator Mathematica.
|
|
file: LliLopMor2014b.preprint.pdf-f630c67919e8d673d0589e74c2adf458.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4104
|
|
bibtexid: GarGiaGinLli2016
|
|
title: Analytic nilpotent centers as limits of nondegenerate centers revisited
|
|
author: García, Isaac A.
|
|
author: Giacomini, Hector
|
|
author: Giné, Jaume
|
|
author: Llibre, Jaume
|
|
journal: Journal of Mathematical Analysis and Applications
|
|
year: 2016
|
|
volume: 441
|
|
startpage: 893
|
|
endpage: 899
|
|
doi: 10.1016/j.jmaa.2016.04.046
|
|
keywords: Nilpotent center
|
|
keywords: Poincaré-Lyapunov constants
|
|
abstract: We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré-Liapunov method to detect centers with purely imaginary eigenvalues can be used to detect nilpotent centers.
|
|
file: GarGiaGinLli2014.preprint.pdf-79870d0efd2b5a7803b070434f962af0.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4106
|
|
bibtexid: CorLliVal2016
|
|
title: Periodic motion in perturbed elliptic oscillators revisited
|
|
author: Corbera, Montserrat
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Astrophysics and Space Science. An International Journal of Astronomy, Astrophysics and Space Science
|
|
year: 2016
|
|
volume: 361:348
|
|
startpage: 1
|
|
endpage: 8
|
|
doi: 10.1007/s10509-016-2927-5
|
|
keywords: Averaging theory
|
|
keywords: galactic potential
|
|
abstract: We analytically study the Hamiltonian system in $\mathbb{R}^4$ with Hamiltonian $$ H= \frac12 (p_x^2 p_y^2) \frac{1}{2} (\omega_1^2 x^2 \omega_2^2 y^2)- \varepsilon\, V_1(x,y) $$ being (a) $V_1(x,y)=-(xy^2 ax^3)$ and (b) $V_1(x,y)=-(x^2y ax^3)$ with $a\in\mathbb{R}$, where $\varepsilon$ is a small parameter and $\omega_1$ and $\omega_2$ are the unperturbed frequencies of the oscillations along the $x$ and $y$ axis, respectively. For the potential (a) using averaging theory of first order we analytically find for each $a\in\mathbb{R}$ eight families of periodic solutions in every positive energy level of $H$ when the frequencies are not equal. For the potential (b) using averaging theory of first and second order we analytically find seven families of periodic solutions in every positive energy level of $H$ when the frequencies are not equal. Four of these seven families are defined for all $a\in\mathbb{R}$ whereas the other three are defined for all $a\ne 0$. Moreover, we provide the shape of all these families of periodic solutions. These Hamiltonians may represent the central parts of deformed galaxies and thus have been extensively used and studied numerically in order to describe local motion in galaxies near an equilibrium point.
|
|
file: CorLliVal2015.Preprint.pdf-6158277bc0a63c12cf5c433737ab3ba9.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4117
|
|
bibtexid: GasGinTor2016
|
|
title: Center problem for systems with two monomial nonlinearities
|
|
author: Gasull, Armengol
|
|
author: Giné, Jaume
|
|
author: Torregrosa, Joan
|
|
journal: Communications on Pure and Applied Analysis
|
|
year: 2016
|
|
volume: 15
|
|
number: 2
|
|
startpage: 577
|
|
endpage: 598
|
|
doi: 10.3934/cpaa.2016.15.577
|
|
keywords: Darboux center
|
|
keywords: Holomorphic center
|
|
keywords: nondegenerate center
|
|
keywords: Persistent center.
|
|
keywords: Poincaré–Lyapunov constants
|
|
keywords: Reversible center
|
|
abstract: We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several centers inside this family. To the best of our knowledge this list includes a new class of Darboux centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the given list is exhaustive. We get several partial results that
|
|
seem to indicate that this is the case. In particular, we solve the question for several general families with arbitrary high degree and for all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest known order for weak-foci of planar polynomial systems of some given degrees.
|
|
file: GasGinTor2014.Preprint.pdf-0ca92203e1ffa3504cdeba898d77c0e4.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4131
|
|
bibtexid: Lli2016
|
|
title: A counterexample to a result on {L}otka-{V}olterra systems
|
|
author: Llibre, Jaume
|
|
journal: Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications
|
|
year: 2016
|
|
volume: 142
|
|
startpage: 123
|
|
endpage: 125
|
|
doi: 10.1007/s10440-015-0019-0
|
|
keywords: Hopf bifurcation
|
|
keywords: Lotka-Volterra system
|
|
abstract: In the article of Dancsó et al. (Acta Appl. Math. 23:103–127, 1991) the authors claim the existence of a Hopf bifurcation which in general does not exist.
|
|
file: Lli2014c.preprint.pdf-bb747ffd388034151b6645c8a511136d.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4145
|
|
bibtexid: GeyMan2016
|
|
title: Singular solutions for a class of traveling wave equations arising in hydrodynamics
|
|
author: Geyer, Anna
|
|
author: Mañosa, Víctor
|
|
journal: Nonlinear Analysis: Real World Applications
|
|
year: 2016
|
|
volume: 31
|
|
startpage: 57
|
|
endpage: 76
|
|
doi: 10.1016/j.nonrwa.2016.01.009 1468-1218/ © 2016
|
|
keywords: Camassa-Holm equation
|
|
keywords: integrable vector fields
|
|
keywords: singular ordinary differential equations
|
|
keywords: traveling waves
|
|
abstract: We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form $\ddot{u}\,u \frac{1}{2}\dot{u}^2 F'(u) =0$, where $F$ is an analytic function. Our motivation stems from the fact that in the context of hydrodynamics several prominent equations are reducible to an equation of this form
|
|
upon passing to a moving frame. We construct peaked and cusped waves, fronts with finite-time decay and compact solitary waves. We prove that one cannot obtain peaked and compactly supported traveling waves for the
|
|
same equation. In particular, a peaked traveling wave cannot have compact support and vice versa. To exemplify the approach we apply our results to the Camassa-Holm equation and the equation for surface waves of moderate amplitude, and show how the different types of singular solutions can be obtained varying the energy level of the corresponding planar Hamiltonian systems.
|
|
file: GeyMan2015.Preprint.pdf-613a41def63c5af366c54788c7566fef.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4150
|
|
bibtexid: LiaTor2016a
|
|
title: Centers of projective vector fields of spatial quasi-homogeneous systems with weight (m,m,n) and degree 2 on the sphere
|
|
author: Liang, Haihua
|
|
author: Torregrosa, Joan
|
|
journal: Electronic Journal of Qualitative Theory of Differential Equations
|
|
year: 2016
|
|
volume: 103
|
|
startpage: 1
|
|
endpage: 26
|
|
doi: 10.14232/ejqtde.2016.1.103
|
|
keywords: projective vector field
|
|
keywords: quasi-homogeneous system
|
|
keywords: sufficient and necessary conditions for centers
|
|
abstract: In this paper we study the centers of projective vector fields $\mathbf{Q}_T$ of three-dimensional quasi-homogeneous differential system $d\mathbf{x}/dt=\mathbf{Q}(\mathbf{x})$ with the weight $(m,m,n)$ and degree $2$ on the unit sphere $\mathbb{S}^2$. We seek the sufficient and necessary conditions under which $\mathbf{Q}_T$ has at least one center on $\mathbb{S}^2$. Moreover, we provide the exact number and the positions of the centers of $\mathbf{Q}_T$. First we give the complete classification of systems $d\mathbf{x}/dt=\mathbf{Q}(\mathbf{x})$ and then, using the induced systems of $\mathbf{Q}_T$ on the local charts of $\mathbb{S}^2,$ we determine the conditions for the existence of centers.
|
|
|
|
The results of this paper provide a convenient criterion to find out all the centers of $\mathbf{Q}_T$ on $\mathbb{S}^2$ with $\mathbf{Q}$ being the quasi-homogeneous polynomial vector field of weight $(m,m,n)$ and degree $2$.
|
|
file: LiangTorregrosa2015a.pdf-6cecfc42679260cae24d063fbcc38b26.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4151
|
|
bibtexid: LiaLliTor2016
|
|
title: Limit cycles coming from some uniform isochronous centers
|
|
author: Liang, Haihua
|
|
author: Llibre, Jaume
|
|
author: Torregrosa, Joan
|
|
journal: Advanced Nonlinear Studies
|
|
year: 2016
|
|
volume: 16
|
|
number: 2
|
|
startpage: 197
|
|
endpage: 220
|
|
doi: 10.1515/ans-2015-5010
|
|
keywords: Averaging theory
|
|
keywords: uniform isochronous centers
|
|
keywords: weak Hilbert problem
|
|
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
|
|
\[
|
|
\dot x= -y x^2 y (x^2 y^2)^n, \dot y= x x y^2 (x^2 y^2)^n,
|
|
\]
|
|
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.
|
|
file: LiaLliTor2016.Preprint.pdf-715fd2f6ef7d837c41852685691b3a46.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4153
|
|
bibtexid: LliSil2016
|
|
title: Global phase portraits of {K}ukles differential systems with homogenous polynomial nonlinearities of degree 5 having a center and their small limit cycles
|
|
author: Llibre, Jaume
|
|
author: Silva, Mauricio Fronza da
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
year: 2016
|
|
volume: 26
|
|
number: 3
|
|
startpage: 1650044 (25 pages)
|
|
doi: 10.1142/S0218127416500449
|
|
keywords: Centers
|
|
keywords: Kukles
|
|
keywords: Phase portrait
|
|
keywords: Poincaré disk
|
|
keywords: Polynomial vector fields
|
|
abstract: We provide the nine topological global phase portraits in the {P}oincaré disk of the family of the centers of Kukles polynomial differential systems of the form $\cdot x = -y$, $\cdot y= x ax^5y bx^3y^3 cxy^5$, where $x,y\in\R$ and $a,b,c$ are real parameters satisfying $a^2 b^2 c^2\neq 0$. Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree 6, and second inside the class of all polynomial differential systems of degree 6.
|
|
file: LliSil2015.Preprint.pdf-b44b3c54348fb081e3d875c1d14f3f82.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4166
|
|
bibtexid: ItiLli2016
|
|
title: Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities
|
|
author: Itikawa, Jackson
|
|
author: Llibre, Jaume
|
|
journal: Discrete and Continuous Dynamical Systems. Series B
|
|
year: 2016
|
|
volume: 21
|
|
number: 1
|
|
startpage: 121
|
|
endpage: 131
|
|
doi: 10.3934/dcdsb.2016.21.121
|
|
keywords: Phase portrait
|
|
keywords: Poincaré disk
|
|
keywords: Polynomial vector field
|
|
keywords: uniform isochronous center
|
|
abstract: We classify the global phase portraits in the Poincar\'{e} disc of the differential systems $\dot{x}=-y xf(x,y),$ $\dot{y}=x yf(x,y)$, where $f(x,y)$ is a homogeneous polynomial of degree 3. These systems have a uniform isochronous center at the origin. This paper together with the results presented in \cite{IL2} completes the classification of the global phase portraits in the Poincar\'{e} disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.
|
|
file: ItiLli2015.Preprint.pdf-c5163ae09fe6565de66363668539b522.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4187
|
|
bibtexid: ManrojVil2016
|
|
title: The criticality of centers of potential systems at the outer boundary
|
|
author: Mañosas, Francesc
|
|
author: Rojas, David
|
|
author: Villadelprat, Jordi
|
|
journal: Journal of Differential Equations
|
|
year: 2016
|
|
volume: 260
|
|
startpage: 4918
|
|
endpage: 4972
|
|
doi: 10.1016/j.jde.2015.11.040
|
|
keywords: Bifurcation
|
|
keywords: Center
|
|
keywords: critical periodic orbit
|
|
keywords: criticality
|
|
keywords: Period function
|
|
abstract: The number of critical periodic orbits that bifurcate from the outer boundary of a potential center is studied. We call this number the criticality at the outer boundary. Our main results provide sufficient conditions in order to ensure that this number is exactly 0 and 1. We apply them to study the bifurcation diagram of the period function of X = −y∂ x ((x 1) p − (x 1) q )∂ y with q < p. This family was previously studied for q = 1 by Y. Miyamoto and K. Yagasaki.
|
|
file: ManRojVil2015.Preprint.pdf-d4114f1263237f3c5a3cadd13eb2ae14.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4195
|
|
bibtexid: CarTor2016
|
|
title: Limit cycles in planar piecewise linear differential systems with nonregular separation line
|
|
author: Cardin, Pedro T.
|
|
author: Torregrosa, Joan
|
|
journal: Physica D. Nonlinear Phenomena
|
|
year: 2016
|
|
volume: 337
|
|
startpage: 67
|
|
endpage: 82
|
|
doi: 10.1016/j.physd.2016.07.008
|
|
keywords: limit cycle in Melnikov higher order perturbation
|
|
keywords: Non-smooth differential systems in two zones
|
|
keywords: nonregular separation line
|
|
abstract: In this paper we deal with lanar piecewise linear differential systems defined in two zones. We consider the case when the two linear zones are angular sectors of angles $\alpha$ and $2\pi - \alpha$, respectively, for $\alpha \in (0,\pi)$. We study the problem of determining lower bounds for the number of isolated periodic orbits in such systems using Melnikov functions. These limit cycles appear studying higher order piecewise linear perturbations of a linear center. It is proved that the maximum number of limit cycles that can appear up to a sixth order perturbation is five. Moreover, for these values of $\alpha$, we prove the existence of systems with four limit cycles up to fifth order and, for $\alpha=\pi/2,$ we provide an explicit example with five up to sixth order. In general, the nonregular separation line increases the number of periodic orbits in comparison with the case where the two zones are separated by a straight line.
|
|
file: CarTor2016.pdf-f153ead5710c7ca4b5ecfe97f74f3cf7.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4210
|
|
bibtexid: RobSilTor2016
|
|
title: Asymptotic expansion of the heteroclinic bifurcation for the planar normal form of the 1:2 resonance
|
|
author: Roberto, Lucy Any
|
|
author: da Silva, Paulo Ricardo
|
|
author: Torregrosa, Joan
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
year: 2016
|
|
volume: 26
|
|
number: 1
|
|
startpage: 1650017 (8 pages)
|
|
doi: 10.1142/S0218127416500176
|
|
keywords: 1:2 Resonance
|
|
keywords: bifurcation diagram
|
|
keywords: Homoclinic Connections
|
|
keywords: Planar Systems
|
|
abstract: We consider the family of planar differential systems depending on two real parameters
|
|
\[\dot {x} =y,\quad \dot {y} = \delta_1 x \delta_2 y x^3-x^2y.\]
|
|
This system corresponds to the normal form for the 1:2 resonance which exhibits a heteroclinic connection. The phase portrait of the system has a limit cycle which disappears in the heteroclinic connection for the parameter values on the curve $\delta_2=c(\delta_1)=-\dfrac{1}{5}\delta_1 O(\delta_1^2),$ $\delta_1<0.$ We significantly improve the knowledge of this curve in a neighborhood of the origin.
|
|
file: RobSilTor2015.Preprint.pdf-3ece9f4f1552d23271e6891c8175612b.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4211
|
|
bibtexid: LiLliWu2016
|
|
title: Polynomial and linearized normal forms for almost periodic differential systems
|
|
author: Li, Weigu
|
|
author: Llibre, Jaume
|
|
author: Wu, Hao
|
|
journal: Discrete and Continuous Dynamical Systems. Series A
|
|
year: 2016
|
|
volume: 36
|
|
number: 1
|
|
startpage: 345
|
|
endpage: 360
|
|
doi: 10.3934/dcds.2016.36.345
|
|
keywords: Almost periodic differential systems
|
|
keywords: Averaging theory
|
|
keywords: linearization
|
|
keywords: Normal form
|
|
file: LiLliWu2016.Preprint.pdf-48e5869f226227ec8c92a3fa8423576d.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4212
|
|
bibtexid: LiaTor2016
|
|
title: Weak-foci of high order and cyclicity
|
|
author: Liang, Haihua
|
|
author: Torregrosa, Joan
|
|
journal: Qualitative Theory of Dynamical Systems
|
|
year: 2016
|
|
doi: 10.1007/s12346-016-0189-9
|
|
keywords: cyclicity
|
|
keywords: Lyapunov quantities
|
|
keywords: polynomial system
|
|
keywords: Weak-focus order
|
|
abstract: A particular version of the 16th Hilbert's problem is to estimate the number, $M(n),$ of limit cycles bifurcating from a singularity of center-focus type. This paper is devoted to finding lower bounds for $M(n)$ for some concrete $n$ by studying the cyclicity of different weak-foci. Since a weak-focus with high order is the most current way to produce high cyclicity, we search for systems with the highest possible weak-focus order. For even $n$, the studied polynomial system of degree $n$ was the one obtained by \cite{QiuYan2009} where the highest weak-focus order is $n^2 n-2$ for $n=4,6,\ldots, 18$. Moreover, we provide a system which has a weak-focus with order $(n-1)^2$ for $n\leq 12$. We show that Christopher's approach \cite{Chr2006}, aiming to study the cyclicity of centers, can be applied also to the weak-focus case. We also show by concrete examples that, in some families, this approach is so powerful and the cyclicity can be obtained in a simple computational way. Finally, using this approach, we obtain that $M(6)\geq 39, M(7)\geq 34$ and $M(8)\geq 63$.
|
|
file: LiaTor2015c.Preprint.pdf-427c1fc0af36bc617bf29063dc4c3eaa.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4238
|
|
bibtexid: GouLliNovPes2016
|
|
title: Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms
|
|
author: Gouveia, Marcio R. A.
|
|
author: Llibre, Jaume
|
|
author: Novaes, Douglas D.
|
|
author: Pessoa, Claudio
|
|
journal: Journal of Differential Equations
|
|
year: 2016
|
|
volume: 260
|
|
startpage: 6108
|
|
endpage: 6129
|
|
doi: 10.1016/j.jde.2015.12.034
|
|
keywords: crossing periodic orbits
|
|
keywords: limit cycle
|
|
keywords: Lyapunov-Schmidt reduction
|
|
keywords: piecewise differential system
|
|
abstract: We consider an n-dimensional piecewise smooth vector field with two zones separated by a hyperplane $\Sigma$ which admits an invariant hyperplane $\Omega$ transversal to $\Sigma$ 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 = 3 we provide normal forms for the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the given normal forms.
|
|
file: GouLliNovPes2016.Preprint.pdf-3a7fc639dcbac1fc67dde36c59ff8dfd.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4266
|
|
bibtexid: LliPan2016
|
|
title: Limit cycles bifurcating from a degenerate center
|
|
author: Llibre, Jaume
|
|
author: Pantazi, Chara
|
|
journal: Mathematics and Computers in Simulation
|
|
year: 2016
|
|
volume: 120
|
|
startpage: 1
|
|
endpage: 11
|
|
doi: 10.1016/j.matcom.2015.05.005
|
|
keywords: Averaging theory
|
|
keywords: Centers
|
|
keywords: limit cycle
|
|
keywords: Polynomial differential systems
|
|
abstract: We study the maximum number of limit cycles that can bifurcate from a degenerate center of a cubic homogeneous polynomial differential system. Using the averaging method of second order and perturbing inside the class of all cubic polynomial differential systems we prove that at most three limit cycles can bifurcate from the degenerate center. As far as we know this is the first time that a complete study up to second order in the small parameter of the perturbation is done for studying the limit cycles which bifurcate from the periodic orbits surrounding a degenerate center (a center whose linear part is identically zero) having neither a Hamiltonian first integral nor a rational one. This study needs many computations, which have been verified with the help of the algebraic manipulator Maple.
|
|
file: LliPan2015.Preprint.pdf-c7a7a1080dd6497eb5fc4faf94337e60.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4267
|
|
bibtexid: DukGinLli2016
|
|
title: Reversible nilpotent centers with cubic homogeneous nonlinearities
|
|
author: Dukarić, Maša
|
|
author: Giné, Jaume
|
|
author: Llibre, Jaume
|
|
journal: Journal of Mathematical Analysis and Applications
|
|
year: 2016
|
|
volume: 433
|
|
startpage: 305
|
|
endpage: 319
|
|
doi: http://dx.doi.org/10.1016/j.jmaa.2015.07.049
|
|
keywords: Nilpotent center
|
|
keywords: Phase portrait
|
|
keywords: Two dimensional differential systems
|
|
abstract: We provide 13 non-topological equivalent classes of global phase portraits in the Poincaré disk of reversible cubic homogeneous systems with a nilpotent center at origin, which complete the classification of the phase portraits of the nilpotent centers with cubic homogeneous nonlinearities.
|
|
file: DukGinLli2016.Preprint.pdf-847c42c7c64210ddd978ef939b441986.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4271
|
|
bibtexid: BusLliGuiVer2016
|
|
title: New families of periodic orbits for a galactic potential
|
|
author: de Bustos, Maria T.
|
|
author: Llibre, Jaume
|
|
author: Guirao, Juan Luis Garcia
|
|
author: Vera, Juan A.
|
|
journal: Chaos, Solitons and Fractals
|
|
year: 2016
|
|
volume: 82
|
|
startpage: 97
|
|
endpage: 102
|
|
doi: 10.1016/j.chaos.2015.11.003
|
|
keywords: Averaging theory
|
|
keywords: family of periodic orbits
|
|
keywords: galactic potential
|
|
abstract: The Hamiltonian system associated to the Hamiltonian
|
|
\[
|
|
H=(P_1^2 P_2^2 P_3^2)/2 (Q_1^2 Q_2^2 Q_3^2)/2 \e(Q_1^4 Q_2^4 Q_3^4 a(Q_1^2Q_2^2 Q_1^2Q_3^2 Q_2^2Q_3^2)),
|
|
\]
|
|
where $\epsilon$ and $a$ are parameters and $\epsilon$ is small, describes the local motion in the central area of a galaxy. Its dynamics have been study by many authors. Here we find analytically new families of periodic orbits of this Hamiltonian system.
|
|
file: BusLliGuiVer2015.Preprint.pdf-e965974a969a5752fceb49d0f8fc3474.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4311
|
|
bibtexid: BraGinLli2016
|
|
title: A sufficient condition in order that the Real {J}acobian Conjecture in {R}^2 holds
|
|
author: Braun, Francisco
|
|
author: Giné, Jaume
|
|
author: Llibre, Jaume
|
|
journal: Journal of Differential Equations
|
|
year: 2016
|
|
volume: 260
|
|
startpage: 5250
|
|
endpage: 5258
|
|
doi: 10.1016/j.jde.2015.12.011
|
|
keywords: centre
|
|
keywords: global injectivity
|
|
keywords: Real Jacobian conjecture
|
|
abstract: Let $F=(f,g):\R^2\to\R^2$ be a polynomial map such that $\det DF(x)$ is different from zero for all $x\in\R^2$ and $F(0,0) = (0,0)$. We prove that for the injectivity of $F$ it is sufficient
|
|
to assume that the higher homogeneous terms of the polynomials $ff_x g g_x$ and $f f_y g g_y$ do not have real linear factors in common.
|
|
file: BraGinLli2015.Preprint.pdf-cb0641c4007ff1c869087546e23bcf97.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4316
|
|
bibtexid: TigLli2016
|
|
title: Heteroclinic, homoclinic and closed orbits in the {C}hen system
|
|
author: Tigan, Gheorghe
|
|
author: Llibre, Jaume
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
year: 2016
|
|
volume: 26
|
|
number: 4
|
|
startpage: 1650072
|
|
doi: 10.1142/S0218127416500723
|
|
keywords: bifurcations
|
|
keywords: homoclinic and heteroclinic orbits
|
|
keywords: ODE systems
|
|
abstract: Bounded orbits such as closed, homoclinic and heteroclinic orbits are discussed in this work for a Lorenz-
|
|
like 3D nonlinear system. For a large spectrum of the parameters the system has neither closed nor homoclinic orbits but has exactly two heteroclinic orbits, while under other constraints the system has symmetrical homoclinic orbits.
|
|
file: TigLli2015.Preprint.pdf-9c662aee138c1e659c277733f5643a4f.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4317
|
|
bibtexid: CimGasMan2016
|
|
title: Periods of solutions of periodic differential equations
|
|
author: Cima, Anna
|
|
author: Gasull, Armengol
|
|
author: Mañosas, Francesc
|
|
journal: Differential and Integral Equations. An International Journal for Theory & Applications
|
|
year: 2016
|
|
volume: 29
|
|
number: 9-10
|
|
startpage: 905
|
|
endpage: 922
|
|
keywords: holomorphic differential equations
|
|
keywords: periodic differential equations
|
|
keywords: periodic orbit
|
|
abstract: Smooth non-autonomous $T$-periodic differential equations $x'(t)=f(t,x(t))$ defined in
|
|
$\R\times\K^n$, where $\K$ is $\R$ or $\C$ and $n\ge 2$ can have periodic solutions with any
|
|
arbitrary period~$S$. We show that this is not the case when $n=1.$ We prove that in the real
|
|
$\mathcal{C}^1$-setting the period of a non-constant periodic solution of the scalar differential
|
|
equation is a divisor of the period of the equation, that is $T/S\in\N.$ Moreover, we
|
|
characterize the structure of the set of the periods of all the periodic solutions of a given
|
|
equation. We also prove similar results in the one-dimensional holomorphic setting. In this
|
|
situation the period of any non-constant periodic solution is commensurable with the period of
|
|
the equation, that is $T/S\in\Q.$
|
|
file: CimGasMan2015a.Preprint.pdf-cddfdf8383643a13ec4a64a7b91cb6c9.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4318
|
|
bibtexid: BraLliMer2016
|
|
title: Isochronicity for trivial quintic and septic planar polynomial {H}amiltonian systems
|
|
author: Braun, Francisco
|
|
author: Llibre, Jaume
|
|
author: Mereu, Ana Cristina
|
|
journal: Discrete and Continuous Dynamical Systems. Series A
|
|
year: 2016
|
|
volume: 36
|
|
number: 10
|
|
startpage: 5245
|
|
endpage: 5255
|
|
doi: 10.3934/dcds.2016029
|
|
keywords: Isochronous centers
|
|
keywords: Jacobian conjecture
|
|
keywords: polynomial Hamiltonian systems
|
|
abstract: In this paper we completely characterize trivial isochronous centers of degrees $5$ and $7$. Precisely, we provide formulas, up to linear change of coordinates, for the Hamiltonian $H$ of the isochronous centers such that $H =(f_1^2 f_2^2)/2$ has degrees $6$ and $8$, and $f = (f_1, f_2): R^2\to R^2$ is a polynomial map with $\det D f = 1$ and $f(0,0) = (0,0)$.
|
|
file: BraLliMer2015.Preprint.pdf-deba1fba9086f3c84e543925666156b0.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4319
|
|
bibtexid: CanLli2016a
|
|
title: New results on averaging theory and applications
|
|
author: Cândido, Murilo R.
|
|
author: Llibre, Jaume
|
|
journal: ZAMP. Journal of Applied Mathematics and Physics
|
|
year: 2016
|
|
volume: 67:106
|
|
startpage: 11p.
|
|
doi: 10.1007/s00033-016-0682-7
|
|
keywords: Averaging theory
|
|
keywords: Fitzhugh--Nagumo system
|
|
keywords: Lorenz system
|
|
keywords: polynomial differential system
|
|
abstract: The usual averaging theory reduces the computation of some periodic solutions of a system of ordinary differential equations, to find the simple zeros of an associated averaged function. When one of these zeros is not simple, i.e. the Jacobian of the averaged function in it is zero, the classical averaging theory does not provide information about the periodic solution associated to a non simple zero. Here we provide sufficient conditions in order that the averaging theory can be applied also to non simple zeros for studying their associated periodic solutions. Additionally we do two applications of this new result for studying the zero--Hopf bifurcation in the Lorenz system and in the Fitzhugh--Nagumo system.
|
|
file: CanLli2015c.Preprint.pdf-6bba1907278aca45cb40087ad2c84e2a.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4322
|
|
bibtexid: GarLliMaz2016
|
|
title: Center cyclicity of a family of quartic polynomial differential system
|
|
author: García, Isaac A.
|
|
author: Llibre, Jaume
|
|
author: Maza, Susanna
|
|
journal: NoDEA : Nonlinear Differential Equations and Applications
|
|
year: 2016
|
|
volume: 23
|
|
number: 34
|
|
startpage: 10 pages
|
|
doi: 10.1007/s00030-016-0388-8
|
|
keywords: Algebraic limit cycles
|
|
keywords: Bautin ideal
|
|
keywords: Center
|
|
keywords: cyclicity
|
|
keywords: Polynomial vector fields
|
|
abstract: In this paper we study the cyclicity of the centers of the quartic polynomial family written in complex notation as
|
|
\[\dot{z} = i z z \bar{z} (A z^2 B z \bar{z} C \bar{z}^2 ),\]
|
|
where $A,B,C \in \mathbb{C}$. We give an upper bound for the cyclicity of any nonlinear center at the origin when we perturb it inside this family. Moreover we prove that this upper bound is sharp.
|
|
file: GarLliMaz2015.Preprint.pdf-7eac23dd52604f3d158bc04b7ca83ddc.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4324
|
|
bibtexid: Lli2016a
|
|
title: Centers: their integrability and relations with the divergence
|
|
author: Llibre, Jaume
|
|
journal: Applied Mathematics and Nonlinear Sciences
|
|
year: 2016
|
|
volume: 1
|
|
number: 1
|
|
startpage: 79
|
|
endpage: 86
|
|
doi: 10.21042/AMNS.2016.1.00007
|
|
keywords: Center problem
|
|
keywords: divergence
|
|
keywords: integrability
|
|
keywords: Poincar\'e--Liapunov constants
|
|
abstract: This is a brief survey on the centers of the analytic differential systems in $\mathbb {R}^2$. First we consider the kind of integrability of the different types of centers, and after we analyze the focus--center problem, i.e. how to distinguish a center from a focus. This is a difficult problem which is not completely solved. We shall present some recent results using the divergence of the
|
|
differential system.
|
|
file: Lli2015b.Preprint.pdf-e55c7f11b77e6aaf00edb45a9ad83259.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4328
|
|
bibtexid: LliZha2016
|
|
title: Limit cycles of linear vectors on manifolds
|
|
author: Llibre, Jaume
|
|
author: Zhang, Xiang
|
|
journal: Nonlinearity
|
|
year: 2016
|
|
volume: 29
|
|
startpage: 3120
|
|
endpage: 3131
|
|
doi: 10.1088/0951-7715/29/10/3120
|
|
keywords: averaging method
|
|
keywords: Center
|
|
keywords: Isochronous center
|
|
keywords: limit cycle
|
|
keywords: periodic orbit
|
|
abstract: It is well known that linear vector fields on the manifold $R^n$ cannot have limit cycles, but this is not the case for linear vector fields on other manifolds. We study the periodic orbits of linear vector fields on different manifolds, and motivate and present an open problem on the number of limit cycles of linear vector fields on a class of $C^1$ connected manifold.
|
|
file: LliZha2015.Preprint.pdf-74f06358733d3c20e9c52922c6165821.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4331
|
|
bibtexid: GasLloMan2015
|
|
title: Continua of periodic points for planar integrable rational maps
|
|
author: Gasull, Armengol
|
|
author: Llorens, Mireia
|
|
author: Mañosa, Víctor
|
|
journal: International Journal of Difference Equations
|
|
year: 2016
|
|
volume: 11
|
|
number: 1
|
|
startpage: 37
|
|
endpage: 63
|
|
keywords: birational maps
|
|
keywords: Integrable rational maps
|
|
keywords: periodic orbits
|
|
abstract: We present three alternative methodologies to find continua of
|
|
periodic points with a prescribed period for rational maps having
|
|
rational first integrals. The first two have been already used for
|
|
other authors and apply when the maps are birational and the
|
|
generic level sets of the corresponding first integrals have either
|
|
genus 0 or 1. As far as we know, the third one is new and it works
|
|
for rational maps without imposing topological properties to the
|
|
invariant level sets. It is based on a computational point of view,
|
|
and relies on the use of resultants in a suitable setting. We apply
|
|
them to several examples, including the 2-periodic Lyness
|
|
composition maps and some of the celebrated McMillan-Gumowski-Mira
|
|
maps.
|
|
file: GasLloMan2015.Preprint.pdf-700baa676e6ddeb13d5f4ecf7db32bfd.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4332
|
|
bibtexid: LliVal2016b
|
|
title: On the analytic integrability of the Liénard analytic differential systems
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Discrete and Continuous Dynamical Systems. Series B
|
|
year: 2016
|
|
volume: 21
|
|
number: 2
|
|
startpage: 557
|
|
endpage: 573
|
|
doi: 10.3934/dcdsb.2016.21.557
|
|
keywords: analytic integrability
|
|
keywords: Formal integrability
|
|
keywords: Li\'{e}nard system
|
|
abstract: We consider the Li\'{e}nard analytic differential systems $\dot x = y$, $\dot y =-g(x) -f(x)y$, where $f,g: \mathbb {R} \to \mathbb {R}$ are analytic functions and the origin is an isolated singular point. Then for such systems we characterize the existence of local analytic first integrals in a neighborhood of the origin and the existence of global analytic first integrals.
|
|
file: LliVal2016b.Preprint.pdf-08e6edec320aaf2c995c7335d06c5239.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4345
|
|
bibtexid: GinLliWuZha2016
|
|
title: Averaging methods of arbitrary order, periodic solutions and integrability
|
|
author: Giné, Jaume
|
|
author: Llibre, Jaume
|
|
author: Wu, Kesheng
|
|
author: Zhang, Xiang
|
|
journal: Journal of Differential Equations
|
|
year: 2016
|
|
volume: 260
|
|
startpage: 4130
|
|
endpage: 4156
|
|
doi: 10.1016/j.jde.2015.11.005
|
|
keywords: averaging method
|
|
keywords: Differential systems
|
|
keywords: integrability
|
|
keywords: limit cycle
|
|
keywords: Polynomial differential systems
|
|
abstract: In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential systems, and it unifies many different kinds of averaging methods. Applying our theory to autonomous analytic differential systems, we obtain some conditions on the existence of limit cycles and integrability. For polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues, we prove that there always exists a positive number N such that if its first N averaging functions vanish, then all averaging functions vanish, and consequently there exists a neighborhood of the origin filled with periodic orbits. Consequently if all averaging functions vanish, the origin is a center for n = 2. Furthermore, in a punctured neighborhood of the origin, the system is $C^\infty$ completely integrable for n > 2 provided that each periodic orbit has a trivial holonomy. Finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities.
|
|
file: GinLliWuZha2016.Preprint.pdf-213695c57efea43381c7f70f414b5755.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4361
|
|
bibtexid: GasGeyMan2016
|
|
title: On the number of limit cycles for perturbed pendulum equations
|
|
author: Gasull, Armengol
|
|
author: Geyer, Anna
|
|
author: Mañosas, Francesc
|
|
journal: Journal of Differential Equations
|
|
year: 2016
|
|
volume: 261
|
|
number: 3
|
|
startpage: 2141
|
|
endpage: 2167
|
|
doi: 10.1016/j.jde.2016.04.025
|
|
keywords: Abelian integrals
|
|
keywords: Infinitesimal Sixteenth Hilbert problem
|
|
keywords: limit cycles
|
|
keywords: Perturbed pendulum equation
|
|
abstract: We consider perturbed pendulum-like equations on the cylinder of the form $ \ddot x \sin(x)= \varepsilon \sum_{\s=0}^{m}{Q_{n,\s} (x)\, \dot x^{\s}}$ where $Q_{n,\s}$ are trigonometric polynomials of degree $n$, and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case $\varepsilon=0$ in terms of $m$ and $n$. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. Some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be Chebyshev systems.
|
|
file: GasGeyMan2016.Preprint.pdf-860fcde78c91533201dd7f659bd02a3b.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4363
|
|
bibtexid: DiaLliMel2016
|
|
title: When parallels and meridians are limit cycles for polynomial vector fields on quadrics of revolution in the euclidean 3-space
|
|
author: Dias, Fabio Scalco
|
|
author: Llibre, Jaume
|
|
author: Mello, Luis Fernando
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
year: 2016
|
|
volume: 26
|
|
number: 9
|
|
startpage: 1650160 (14 pages)
|
|
doi: 10.1142/S0218127416501601
|
|
keywords: invariant meridian
|
|
keywords: invariant parallel
|
|
keywords: limit cycle
|
|
keywords: periodic orbit
|
|
keywords: Polynomial vector field
|
|
abstract: We study polynomial vector fields of arbitrary degree in ${R}^3$ with an invariant quadric of revolution. We characterize all the possible configurations of invariant meridians and parallels that these vector fields can exhibit. Furthermore we analyze when these invariant meridians and parallels can be limit cycles.
|
|
file: DiaLliMel2016.preprint.pdf-b9a9e6f7035353b6aa6e5e1e8979f3e2.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4368
|
|
bibtexid: GasTorZha2016
|
|
title: The number of polynomial solutions of polynomial {R}iccati equations
|
|
author: Gasull, Armengol
|
|
author: Torregrosa, Joan
|
|
author: Zhang, Xiang
|
|
journal: Journal of Differential Equations
|
|
year: 2016
|
|
volume: 261
|
|
startpage: 5071
|
|
endpage: 5093
|
|
doi: 10.1016/j.jde.2016.07.019
|
|
keywords: explicit solutions
|
|
keywords: number of polynomial solutions
|
|
keywords: Polynomial differential equations
|
|
keywords: Riccati differential equations
|
|
keywords: trigonometric polynomial differential equations
|
|
abstract: Consider real or complex polynomial Riccati differential equations $a(x) \dot y=b_0(x) b_1(x)y b_2(x)y^2$ with all the involved functions being polynomials of degree at most $\eta$. We prove that the maximum number of polynomial solutions is $\eta 1$ (resp. 2) when $\eta\ge 1$ (resp. $\eta=0$) and that these bounds are sharp.
|
|
|
|
For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most $\eta\ge 1$ we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is $2\eta$ (resp. $3$) when $\eta\ge 2$ (resp. $\eta=1$) and, again, these bounds are sharp.
|
|
|
|
Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.
|
|
file: GasTorZha2016.Preprint.pdf-6605adc2f582d8c93dc0789942e9aaaf.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4377
|
|
bibtexid: GinLli2016
|
|
title: Canards Existence in {M}emristor’s Circuits
|
|
author: Ginoux, Jean-Marc
|
|
author: Llibre, Jaume
|
|
journal: Qualitative Theory of Dynamical Systems
|
|
year: 2016
|
|
doi: 10.1007/s12346-015-0160-1
|
|
keywords: canard solutions
|
|
keywords: Geometric singular perturbation theory
|
|
keywords: singularly perturbed dynamical systems
|
|
abstract: The aim of this work is to propose an alternative method for determining the condition of existence of “canard solutions” for three and four-dimensional singularly perturbed systems with only one fast variable in the folded saddle case. This method enables to state a unique generic condition for the existence of “canard solutions”
|
|
for such three and four-dimensional singularly perturbed systems which is based on the stability of folded singularities of the normalized slow dynamics deduced from a well-known property of linear algebra. This unique generic condition is perfectly identical to that provided in previous works. Application of this method to the famous
|
|
three and four-dimensional memristor canonical Chua’s circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to show the existence of “canard solutions” in such Memristor Based Chaotic Circuits.
|
|
file: GinLli2016.Preprint.pdf-a6e9df675365ff2aa99a640a913ffbdc.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4383
|
|
bibtexid: CanFagGar2016
|
|
title: Tongues in Degree 4 {B}laschke Products
|
|
author: Canela, Jordi
|
|
author: Fagella, Nuria
|
|
author: Garijo, Antoni
|
|
journal: Nonlinearity
|
|
year: 2016
|
|
volume: 29
|
|
startpage: 3464
|
|
endpage: 3495
|
|
doi: 10.1088/0951-7715/29/11/3464
|
|
keywords: Blaschke products
|
|
keywords: circle maps
|
|
keywords: Holomorphic dynamics
|
|
keywords: tongues
|
|
abstract: The goal of this paper is to investigate the family of Blasche products $B_a(z)=z^3\frac{z-a}{1-\bar{a}z}$, which is a rational family of perturbations of the doubling map. We focus on the tongue-like sets which appear in its parameter plane. We first study their basic topological properties and afterwords we investigate how bifurcations take place in a neighborhood of their tips. Finally we see how the period one tongue extends beyond its natural domain of definition.
|
|
file: CanFagGar2016.preprint.pdf-bd18cbb0a111c36b82848aa919e1e360.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4388
|
|
bibtexid: AlsManMor2016
|
|
title: A quasiperiodically forced skew-product on the cyclinder without fixed-curves
|
|
author: Alsedà, Lluís
|
|
author: Mañosas, Francesc
|
|
author: Morales, Leopoldo
|
|
journal: Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods
|
|
year: 2016
|
|
volume: 145
|
|
startpage: 199
|
|
endpage: 263
|
|
keywords: invariant strips
|
|
keywords: Quasiperiodically forced systems on the cylinder
|
|
abstract: In [FJJK] the Sharkovski\uı Theorem was extended to periodic orbits of strips of quasiperiodic skew products in the cylinder. In this paper we deal with the following natural question that arises in this setting: Does Sharkovski\uı Theorem holds when restricted to curves instead of general strips? We answer this question in the negative by constructing a counterexample: We construct a map having a periodic orbit of period 2
|
|
of curves (which is, in fact, the upper and lower circles of the cylinder) and without any invariant curve. In particular this shows that there exist quasiperiodic skew products in the cylinder without invariant curves.
|
|
|
|
[FJJK] Roberta Fabbri, Tobias Jäger, Russell Johnson, and Gerhard Keller. A {S}harkovskii-type theorem for minimally forced interval maps. Topol. Methods Nonlinear Anal., 26(1):163--188, 2005.
|
|
file: AlsManMor2016.Preprint.pdf-87ff42b7a5d80dbdc1cc0682322dd765.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4390
|
|
bibtexid: CamGarJarVin2016
|
|
title: Newton's method for symmetric quartic polynomials
|
|
author: Campos, Beatriz
|
|
author: Garijo, Antoni
|
|
author: Jarque, Xavier
|
|
author: Vindel, Pura
|
|
journal: Applied Mathematics and Computation
|
|
year: 2016
|
|
volume: 290
|
|
startpage: 326
|
|
endpage: 335
|
|
doi: 10.1016/j.amc.2016.06.021
|
|
keywords: Holomorphic dynamics
|
|
keywords: Julia and Fatou sets
|
|
keywords: Newton’s method
|
|
abstract: We investigate the parameter plane of the Newton's method applied to the family of quartic polynomials
|
|
$p_{a,b}(z)=z^4 az^3 bz^2 az 1$, where $a$ and $b$ are real parameters. We divide the parameter plane
|
|
$(a,b) \in \mathbb R^2$ into twelve open and connected {\it regions} where $p$, $p'$ and $p''$ have
|
|
simple roots. In each of these regions we focus on the study of the Newton's operator acting on the Riemann sphere.
|
|
file: CAMGARJARVIN2016.pdf-ab9aec38243579caca6b620f7d00cec2.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4402
|
|
bibtexid: GuiLli2016a
|
|
title: Periods of continuous maps on some compact spaces
|
|
author: Guirao, Juan Luis Garcia
|
|
author: Llibre, Jaume
|
|
journal: Houston Journal of Mathematics
|
|
year: 2016
|
|
volume: 42
|
|
number: 3
|
|
startpage: 1047
|
|
endpage: 1058
|
|
keywords: complex projective space
|
|
keywords: continuous map
|
|
keywords: Lefschetz fixed point theory
|
|
keywords: periodic point
|
|
keywords: Periods
|
|
keywords: product of two spheres
|
|
keywords: quaternion projective space
|
|
keywords: sphere
|
|
abstract: The objective of this paper is to provide information on the set of periodic points of a continuous self--map defined in the following compact spaces: $\mathbb{S}^{n}$ (the $n$--dimensional sphere), $\mathbb{S}^{n}\times \mathbb{S}^{m}$ (the product space of the $n$--dimensional with the $m$--dimensional spheres), $\mathbb{C}P^{n}$ (the $n$--dimensional complex projective space) and $\mathbb{H}P^{n}$ (the $n$--dimensional quaternion projective space). We use as main tool the action of the map on the homology groups of these compact spaces.
|
|
file: GuiLli2016a.preprint.pdf-4fb13ac063abea42ab66d5c85081f2c6.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Phdthesis
|
|
aigaionid: 4410
|
|
bibtexid: Mor2016
|
|
title: Combinatorial dynamics of strip patterns of quasiperiodic skew products in the cylinder
|
|
author: Morales, Leopoldo
|
|
year: 2016
|
|
school: Universitat Autònoma de Barcelona
|
|
address: Bellaterra
|
|
file: Mor2016.pdf-d9e415f7c08d549434ad7cbbf4d51cb3.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Phdthesis
|
|
aigaionid: 4415
|
|
bibtexid: Roj2016
|
|
title: Analytical tools to study the criticality at the outer boundary of potential centers
|
|
author: Rojas, David
|
|
year: 2016
|
|
school: Universitat Autònoma de Barcelona
|
|
address: Bellaterra
|
|
file: Roj2016.pdf-413201363d3d731a45afd08bc9b2fdef.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4426
|
|
bibtexid: BujLliVul2016
|
|
title: First integrals and phase portraits of planar polynomial differential cubic systems with the maximum number of invariant straight lines
|
|
author: Bujac, Cristina
|
|
author: Llibre, Jaume
|
|
author: Vulpe, Nicolae
|
|
journal: Qualitative Theory of Dynamical Systems
|
|
year: 2016
|
|
volume: 15
|
|
startpage: 327
|
|
endpage: 348
|
|
abstract: In the article LliVul2006 the family of cubic polynomial
|
|
differential systems possessing invariant straight lines of total
|
|
multiplicity 9 was considered and 23 such classes of systems were
|
|
detected. We recall that 9 invariant straight lines taking into
|
|
account their multiplicities is the maximum number of straight
|
|
lines that a cubic polynomial differential systems can have if
|
|
this number is finite. Here we complete the classification given
|
|
in LliVul2006 by adding a new class of such cubic
|
|
systems and for each one of these 24 such classes we perform the
|
|
corresponding first integral as well as its phase portrait.
|
|
Moreover we present necessary and sufficient affine invariant
|
|
conditions for the realization of each one of the detected classes
|
|
of cubic systems with maximum number of invariant straight lines
|
|
when this number is finite.
|
|
file: BujLliVul2016.preprint.pdf-150014e6d7de07650cee7089931f2903.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4427
|
|
bibtexid: GinLli2016d
|
|
title: Analytic reducibility of nondegenerate centers: {C}herkas systems
|
|
author: Giné, Jaume
|
|
author: Llibre, Jaume
|
|
journal: Electronic Journal of Qualitative Theory of Differential Equations
|
|
year: 2016
|
|
volume: 49
|
|
startpage: 1
|
|
endpage: 10
|
|
doi: doi: 10.14232/ejqtde.2016.1.49
|
|
keywords: analytic integrability
|
|
keywords: Center problem
|
|
keywords: polynomial Cherkas differential systems
|
|
file: GinLli2016d.preprint.pdf-2a85970dc88a865ec8937a739c65a69c.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4436
|
|
bibtexid: LliVid2016
|
|
title: Hopf periodic orbits for a ratio-dependent predator-prey model with stage structure
|
|
author: Llibre, Jaume
|
|
author: Vidal, Claudio
|
|
journal: Discrete and Continuous Dynamical Systems. Series B
|
|
year: 2016
|
|
volume: 21
|
|
number: 6
|
|
startpage: 1859
|
|
endpage: 1867
|
|
doi: 10.3934/dcdsb.2016026
|
|
keywords: Averaging theory
|
|
keywords: Hopf bifurcation
|
|
keywords: predator-prey model
|
|
keywords: Ratio–dependence
|
|
abstract: A ratio–dependent predator-prey model with stage structure for prey was investigated in [8]. There the authors mentioned that they were unable to show if such a model admits limit cycles when the unique equilibrium
|
|
point E ∗ at the positive octant is unstable. Here we characterize the existence of Hopf bifurcations for the systems. In particular we provide a positive answer to the above question showing for such models the existence of small–amplitude Hopf limit cycles being the equilibrium point E ∗ unstable.
|
|
file: LliVid2016.preprint.pdf-4a5e2c879f1585724c566f0cfe16fdc6.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4444
|
|
bibtexid: LemLli2016a
|
|
title: Periodic orbits of perturbed elliptic oscillators in 6{D} via averaging theory
|
|
author: Lembarki, Fatima E.
|
|
author: Llibre, Jaume
|
|
journal: Astrophysics and Space Science. An International Journal of Astronomy, Astrophysics and Space Science
|
|
year: 2016
|
|
startpage: 361
|
|
endpage: 340
|
|
doi: 10.1007/s10509-016-2930-x
|
|
keywords: Averaging theory
|
|
keywords: galactic dynamics
|
|
keywords: periodic orbits
|
|
keywords: perturbed elliptic oscillators
|
|
abstract: We provide sufficient conditions on the energy levels to guarantee
|
|
the existence of periodic orbits for the perturbed elliptic
|
|
oscillators in 6D using the averaging theory. We give also an
|
|
analytical estimation of the shape of these periodic orbits
|
|
parameterized by the energy. The Hamiltonian system here studied
|
|
comes either from the analyze of the galactic dynamics, or from the
|
|
motion of the atomic particles in physics.
|
|
file: LemLli2016a.preprint.pdf-c81eeb200fb545fba26546e90c030b0e.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
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aigaionid: 4471
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bibtexid: OsRebVi2016
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title: On a Class of Invariant Algebraic Curves for {K}ukles Systems
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author: Osuna, Osvaldo
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author: Rebollo-Perdomo, Salomón
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author: Villaseñor, Gabriel
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journal: Electronic Journal of Qualitative Theory of Differential Equations
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year: 2016
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volume: 2016
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number: 61
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startpage: 1
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endpage: 12
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doi: 10.14232/ejqtde.2016.1.61
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keywords: invariant curve
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keywords: Kukles system
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file: OsRebVi2016.Preprint.pdf-e84e4789358c014f4e02cba9e622174d.pdf
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</entry>
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<entry>
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type: Article
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aigaionid: 4487
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bibtexid: ArtOliRez2016
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title: Topological Classification of Quadratic Polynomial Differential Systems with a Finite Semi-Elemental Triple Saddle
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author: Artés, Joan Carles
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author: Oliveira, Regilene D. S.
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author: Rezende, Alex Carlucci
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journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
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year: 2016
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volume: 26
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number: 11
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startpage: 1650188 (26 pages)
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doi: 10.1142/S0218127416501881
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keywords: algebraic invariants
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keywords: bifurca- tion diagram
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keywords: phase portraits
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keywords: Quadratic differential systems
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keywords: semi-elemental triple saddle
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abstract: The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilbert’s 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family QTS of all real quadratic polynomial differential systems which have a finite semi-elemental triple saddle (triple saddle with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this normal form. This bifur- cation diagram yields 27 phase portraits for systems in QTS counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set and we present the phase portraits on the Poincar ́e disk. The bifurcation set is not just algebraic due to the presence of a surface found numerically, whose points correspond to connections of separatrices.
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file: ArtOliRez2016.preprint.pdf-e2b019d666cd1a6f8219d7ebfb9553b1.pdf
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</entry>
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<entry>
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type: Article
|
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aigaionid: 3652
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bibtexid: AlsJuhMan2017
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title: On the minimum positive entropy for cycles on trees
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author: Alsedà, Lluís
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author: Juher, David
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author: Mañosas, Francesc
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journal: Transactions of the American Mathematical Society
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year: 2017
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volume: 369
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|
number: 1
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startpage: 187
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|
endpage: 221
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|
file: AlsJuhMan2013.pdf-e8f2700425b79d14a2d915c9ad096a60.pdf
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</entry>
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<entry>
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type: Article
|
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aigaionid: 4330
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|
bibtexid: BenLli2017
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|
title: Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory
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author: Benterki, Rebiha
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author: Llibre, Jaume
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journal: Journal of Computational and Applied Mathematics
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year: 2017
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volume: 313
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|
startpage: 273
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|
endpage: 283
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|
doi: 10.1016/j.cam.2016.08.047
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keywords: averaging method
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keywords: Center
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keywords: generalized Kukles system
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keywords: limit cycle
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keywords: Phase portrait
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abstract: In this paper we classify the phase portraits in the Poincar\'{e} disc of the centers of the generalized class of Kukles systems \[ \dot{x}=-y,\quad\dot{y}=x ax^3y bxy^3, \] symmetric with respect to the $y$-axis, and we study, using the averaging theory up to sixth order, the limit cycles which bifurcate from the periodic solutions of these centers when we perturb them inside the class of all polynomial differential systems of degree $4$.
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file: BenLli2015.Preprint.pdf-4de0dae45f675df249a2b3ed575db59d.pdf
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|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4393
|
|
bibtexid: ArtItiLli2017
|
|
title: Uniform isochronous cubic and quartic centers: {R}evisited
|
|
author: Artés, Joan Carles
|
|
author: Itikawa, Jackson
|
|
author: Llibre, Jaume
|
|
journal: Journal of Computational and Applied Mathematics
|
|
year: 2017
|
|
volume: 313
|
|
startpage: 448
|
|
endpage: 453
|
|
keywords: periodic orbit
|
|
keywords: Polynomial vector field
|
|
keywords: uniform isochronous center
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|
abstract: In this paper we completed the classification of the phase portraits in the Poincaré disc of uniform isochronous cubic and quartic centers previously studied by several authors. There are three and fourteen different topological phase portraits for the uniform isochronous cubic and quartic centers respectively.
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file: ArtItiLli2016.preprint.pdf-8b4be3671a1e4bf72fcb2804d30fe952.pdf
|
|
</entry>
|
|
<entry>
|
|
type: Article
|
|
aigaionid: 4435
|
|
bibtexid: BuzPazPer2017
|
|
title: Center boundaries for planar piecewise-smooth differential equations with two zones
|
|
author: Buzzi, Claudio Aguinaldo
|
|
author: Pazim, Rubens
|
|
author: Pérez-González, Set
|
|
journal: Journal of Mathematical Analysis and Applications
|
|
year: 2017
|
|
volume: 445
|
|
startpage: 631
|
|
endpage: 649
|
|
doi: 10.1016/j.jmaa.2016.07.022
|
|
keywords: limit cycle
|
|
keywords: non-smooth differential system
|
|
keywords: Piecewise linear differential system
|
|
abstract: This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for
|
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a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary. We prove that given a pair of systems that share a hyperbolic focus singularity p 0 , with the same orientation and opposite stability, and a ray Σ 0 with endpoint at the singularity p 0 , we can find a smooth manifold Ω such that Σ 0 ∪ {p 0 } ∪ Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached.
|
|
file: BuzPazPer2016.Preprint.pdf-0344f3e8ae0d57bf70d7fe1b9afbfb19.pdf
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|
</entry>
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