|
type: Article
|
|
aigaionid: 4703
|
|
bibtexid: LliYu2018
|
|
title: Global phase portraits for quadratic systems with a hyperbola and a straight line as invariant algebraic curves
|
|
author: Llibre, Jaume
|
|
author: Yu, Jiang
|
|
journal: Electronic Journal of Differential Equations
|
|
issn: 1072-6691
|
|
year: 2018
|
|
number: 141
|
|
startpage: 1
|
|
endpage: 19
|
|
abstract: In this article we consider a class of quadratic polynomial differential systems in the plane having a hyperbola and a straight line as invariant algebraic curves, and we classify all its phase portraits. Moreover these systems are integrable and we provide their first integrals.
|
|
file: LliYu2018.Preprint.pdf-27eb2e4766a4413565302d0b6a2a7faf.pdf
|
|
|
|
type: Article
|
|
aigaionid: 4704
|
|
bibtexid: LliZha2018c
|
|
title: Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center
|
|
author: Llibre, Jaume
|
|
author: Zhang, Xiang
|
|
journal: Journal of Mathematical Analysis and Applications
|
|
issn: 0022-247X
|
|
year: 2018
|
|
volume: 467
|
|
startpage: 537
|
|
endpage: 549
|
|
doi: 10.1016/j.jmaa.2018.07.024
|
|
keywords: discontinuous piecewise linear differential system
|
|
keywords: limit cycle
|
|
keywords: non-smooth differential system
|
|
abstract: From the beginning of this century more than thirty papers have been published studying the limit cycles of the discontinuous piecewise linear differential systems with two pieces separated by a straight line, but it remains open the following question: what is the maximum number of limit cycles that this class of differential systems can have? Here we prove that when one of the linear differential systems has a center, real or virtual, then the discontinuous piecewise linear differential system has at most two limit cycles.
|
|
file: LliZha2018.Preprint.pdf-b8f4184f43fc12ab2221b5e2dcc3d73f.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4520
|
|
bibtexid: MarVil2018
|
|
title: On the {C}hebyshev property of certain {A}belian integrals near a polycycle
|
|
author: Marín, David
|
|
author: Villadelprat, Jordi
|
|
journal: Qualitative Theory of Dynamical Systems
|
|
issn: 1575-5460
|
|
year: 2018
|
|
volume: 17
|
|
number: 1
|
|
startpage: 261
|
|
endpage: 270
|
|
doi: 10.1007/s12346-017-0226-3
|
|
keywords: Abelian integrals
|
|
keywords: Chebyshev system
|
|
keywords: Wronskian
|
|
abstract: F. Dumortier and R. Roussarie formulated in [Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009) 723–781] a conjecture concerning the Chebyshev property of a collection $I_0,I_1,...,I_n$ of Abelian integrals arising from singular perturbation problems occurring in planar slow-fast systems. The aim of this note is to show the validity of this conjecture near the polycycle at the boundary of the family of ovals defining the Abelian integrals. As a corollary of this local result we get that the linear span $⟨I_0, I_1, . . . , I_n⟩$ is Chebyshev with accuracy $k = k(n)$.
|
|
file: MarVil2017.preprint.pdf-0d9b2631a2252483079c9feffe4638be.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 3630
|
|
bibtexid: BenFag2018a
|
|
title: Singular values and bounded {S}iegel disks
|
|
author: Benini, Anna Miriam
|
|
author: Fagella, Nuria
|
|
journal: Mathematical Proceedings of the Cambridge Philosophical Society
|
|
issn: 0305-0041
|
|
year: 2018
|
|
volume: 165
|
|
number: 2
|
|
startpage: 249
|
|
endpage: 265
|
|
doi: 10.1017/S0305004117000469
|
|
abstract: Let f be an entire transcendental function of finite order and ? be a forward invariant bounded Siegel disk for f with rotation number in Herman's class H. We show that if f has two singular values with bounded orbit, then the boundary of ? contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.
|
|
file: BenFag2013.pdf-53eb443c4360b511911033ee50890390.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 3664
|
|
bibtexid: LliVal2018i
|
|
title: On the {D}arboux integrability of the {H}indmarsh-{R}ose burster
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Acta Mathematica Sinica. English Series
|
|
issn: 1439-7617
|
|
year: 2018
|
|
volume: 34
|
|
number: 6
|
|
startpage: 947
|
|
endpage: 958
|
|
doi: 10.1007/s10114-017-5661-1
|
|
abstract: We study the Hindmarsh?Rose burster which can be described by the differential system
|
|
x?=y?x3+bx2+I?z,y?=1?5x2?y,z?=?(s(x?x0)?z),
|
|
where b, I, ?, s, x0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist.
|
|
file: LliVal2013t.pdf-9d0403a2d819ba9fb8ff964e4ba65d74.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4101
|
|
bibtexid: LliOliRod2018
|
|
title: On the periodic solutions of the {M}ilchelson continuous and discontinuous piecewise linear differential system
|
|
author: Llibre, Jaume
|
|
author: Oliveira, Regilene D. S.
|
|
author: Rodrigues, Camila A. B.
|
|
journal: Computational & Applied Mathematics
|
|
issn: 0101-8205
|
|
year: 2018
|
|
volume: 37
|
|
number: 2
|
|
startpage: 1550
|
|
endpage: 1561
|
|
doi: 10.1007/s40314-016-0413-x
|
|
keywords: Averaging theory
|
|
keywords: continuous piecewise linear differential system
|
|
keywords: discontinuous piecewise linear differential system
|
|
keywords: Hopf bifurcation
|
|
keywords: limit cycles
|
|
keywords: Michelson system
|
|
abstract: Applying new results from the averaging theory for continuous and discontinuous differential systems, we study the periodic solutions of two distinct versions of the Michel- son differential system: a Michelson continuous piecewise linear differential system and a Michelson discontinuous piecewise linear differential system. The tools here used can be applied to general nonsmooth differential systems.
|
|
file: LliOliRod2014.preprint.pdf-7d77334f00d23573fe9b84031d2930e6.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4138
|
|
bibtexid: CanLli2018c
|
|
title: Zero-{H}opf bifurcations in a hyperchaotic {L}orenz system {I}{I}
|
|
author: Cândido, Murilo R.
|
|
author: Llibre, Jaume
|
|
journal: International Journal of Nonlinear Science
|
|
issn: 1749-3889
|
|
year: 2018
|
|
volume: 25
|
|
number: 1
|
|
startpage: 3
|
|
endpage: 26
|
|
keywords: Averaging theory
|
|
keywords: hyperchaotic Lorenz system
|
|
keywords: periodic orbit
|
|
keywords: zero-Hopf bifurcation
|
|
abstract: Recently sixteen 3-dimensional differential systems exhibiting chaotic motion and having no equilibria have been studied, and it has been graphically observed that these systems have a period-doubling cascade of periodic orbits providing the route to their chaotic motions. Here using new results on the averaging theory we prove that these systems exhibit, for some values of their parameters different to the ones having chaotic motion, either a zero?Hopf or a Hopf bifurcation, and graphically we observed that the periodic orbit starting in those bifurcations is at the beginning of the mentioned period?doubling cascade.
|
|
file: CanLli2017.preprint.pdf-6432eb4fe40b5eb9240064a302860be4.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4141
|
|
bibtexid: BarFagJarKar2018
|
|
title: Connectivity of {J}ulia sets of {N}ewton maps: A unified approach
|
|
author: Baranski, Krzysztof
|
|
author: Fagella, Nuria
|
|
author: Jarque, Xavier
|
|
author: Karpinska, Boguslawa
|
|
journal: Revista Mathemática Iberoamericana
|
|
issn: 0213-2230
|
|
year: 2018
|
|
volume: 34
|
|
number: 3
|
|
startpage: 1211
|
|
endpage: 1228
|
|
doi: 10.4171/RMI/1022
|
|
keywords: connectivity
|
|
keywords: Fatou set
|
|
keywords: Holomorphic dynamics
|
|
keywords: Julia set
|
|
keywords: Newton's map
|
|
keywords: repelling fixed point
|
|
keywords: simple connectivity
|
|
abstract: In this paper we give a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function of the complex plane (a polynomial of degree large than 1 or an entire transcendental function) is connected. The result was recently completed by the authors' previous work, as a consequence of a more general theorem whose proof spreads among many papers, which consider separately a number of particular cases for rational and transcendental maps, and use a variety of techniques. In this note we present a unified, direct and reasonably self-contained proof which works for all situations alike.
|
|
file: BarFagJarKar2015.pdf-df1653ccdbd345da4ebb0e33fee66bf8.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4321
|
|
bibtexid: LliVal2018b
|
|
title: Normal forms and hyperbolic algebraic limit cycles for a class of polynomial differential systems
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Electronic Journal of Differential Equations
|
|
issn: 1072-6691
|
|
year: 2018
|
|
number: 83
|
|
startpage: 1
|
|
endpage: 7
|
|
keywords: Algebraic limit cycles
|
|
keywords: limit cycles
|
|
keywords: Polynomial vector fields
|
|
abstract: We study the normal forms of polynomial systems having a set of invariant algebraic curves with singular points. We provide sufficient conditions for the existence of hyperbolic algebraic limit cycles.
|
|
file: LliVal2015o.Preprint.pdf-94dbd08cbff2fe8dc0e02dd7e6f84290.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4230
|
|
bibtexid: LliRamRamSad2018
|
|
title: Centers and uniform isochronous centers of planar polynomial differential systems
|
|
author: Llibre, Jaume
|
|
author: Ramírez, Rafael Orlando
|
|
author: Ramírez, Valentín
|
|
author: Sadovskaia, Natalia
|
|
journal: Journal of Dynamics and Differential Equations
|
|
issn: 1572-9222
|
|
year: 2018
|
|
volume: 30
|
|
number: 3
|
|
startpage: 1295
|
|
endpage: 1310
|
|
doi: 10.1007/s10884-018-9672-0
|
|
keywords: Center-focus problem
|
|
keywords: polynomial planar differential system
|
|
keywords: uniform isochronous centers
|
|
abstract: For planar polynomial vector fields of the form \[ (-y X(x,y))\dfrac{\partial }{\partial x} (x Y(x,y))\dfrac{\partial }{\partial y}, \] where $X$ and $Y$ start at least with terms of second order in the variables $x$ and $y$, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers.
|
|
file: LliRamSad2015.Preprint.pdf-3f6d27d52c84c714fb54833395b48cad.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4235
|
|
bibtexid: LliVal2018h
|
|
title: Global phase portraits of quadratic systems with a complex ellipse as invariant algebraic curve
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Acta Mathematica Sinica. English Series
|
|
issn: 1439-7617
|
|
year: 2018
|
|
volume: 34
|
|
number: 5
|
|
startpage: 801
|
|
endpage: 811
|
|
doi: 10.1007/s10114-017-5478-y
|
|
keywords: complex ellipse
|
|
keywords: Invariant algebraic curves
|
|
keywords: Phase portrait
|
|
keywords: quadratic system
|
|
abstract: In this paper we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse $x^2 y^2 1=0$ as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincaré disc.
|
|
file: LliVal2015k.Preprint.pdf-67b5ee36f4f3b4c291afbb0eb2eaab1e.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4313
|
|
bibtexid: GinLliVal2018b
|
|
title: The cubic polynomial differential systems with two circles as algebraic limit cycles
|
|
author: Giné, Jaume
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Advanced Nonlinear Studies
|
|
issn: 2169-0375
|
|
year: 2018
|
|
volume: 18
|
|
number: 1
|
|
startpage: 183
|
|
endpage: 193
|
|
doi: 10.1515/ans-2017-6033
|
|
keywords: Cubic systems
|
|
keywords: global phase portraits
|
|
keywords: Invariant algebraic curves
|
|
keywords: invariant ellipse
|
|
keywords: limit cycles
|
|
abstract: In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.
|
|
file: GinLliVal2015b.Preprint.pdf-af649c11633242c2796a7a5ab39fac84.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4180
|
|
bibtexid: FreLliMed2018
|
|
title: Limit cycles of continuous and discontinuous piecewise-linear differential systems in $\mathbb {R}^3$
|
|
author: de Freitas, Bruno R.
|
|
author: Llibre, Jaume
|
|
author: Medrado, Joao Carlos
|
|
journal: Journal of Computational and Applied Mathematics
|
|
issn: 0377-0427
|
|
year: 2018
|
|
volume: 338
|
|
startpage: 311
|
|
endpage: 323
|
|
doi: 10.1016/j.cam.2018.01.028
|
|
keywords: limit cycles
|
|
keywords: Non-smooth differential systems
|
|
keywords: piecewise linear differential systems
|
|
abstract: We study the limit cycles of two families of piecewise-linear differential systems in $\R^3$ with two pieces separated by a plane $\Sigma$. In one family the differential systems are only continuous on the plane $\Sigma$, and in the other family they are only discontinuous on the plane $\Sigma$. The usual tool for studying these limit cycles is the Poincar\'{e} map, but here we shall use recent results which extend the averaging theory to continuous and discontinuous differential systems. All the computations have been checked with the algebraic manipulator mathematica.
|
|
file: FreLliMed2015.preprint.pdf-b48dac115eddece803d527d94c0b083c.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4209
|
|
bibtexid: LliOli2018
|
|
title: Quadratic systems with an invariant conic having Darboux invariants
|
|
author: Llibre, Jaume
|
|
author: Oliveira, Regilene D. S.
|
|
journal: Communications in Contemporaray Mathematics
|
|
issn: 1793-6683
|
|
year: 2018
|
|
volume: 20
|
|
number: 4
|
|
startpage: 1750033
|
|
doi: 10.1142/S021919971750033X
|
|
keywords: Darboux invariant
|
|
keywords: phase portraits
|
|
keywords: Quadratic vector fields
|
|
abstract: The complete characterization of the phase portraits of real planar quadratic vector fields is very far to be completed. As this attempt is not possible in the whole class due to the large number of parameters (twelve, but, after affine transformations and time rescaling, we arrive at families with five parameters, which is still a big number of parameters), many subclasses have been considered and studied. In this paper we complete the characterization of the global phase portraits in the Poincaré disc of all planar quadratic polynomial differential systems having an invariant conic and a Darboux invariant, constructed using only the invariant conic.
|
|
file: LliReg2015.Preprint.pdf-2b64416dc03080412eddd07cd3dea765.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4582
|
|
bibtexid: HuaLiaLli2018
|
|
title: Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
|
|
author: Huang, Jianfeng
|
|
author: Liang, Haihua
|
|
author: Llibre, Jaume
|
|
journal: Journal of Differential Equations
|
|
issn: 0022-0396
|
|
year: 2018
|
|
volume: 265
|
|
number: 9
|
|
startpage: 3888
|
|
endpage: 3913
|
|
doi: 10.1016/j.jde.2018.05.019
|
|
keywords: homogeneous nonlinearities
|
|
keywords: limit cycles
|
|
keywords: non-existence and uniqueness
|
|
keywords: Polynomial differential systems
|
|
abstract: In this paper we study the limit cycles of the planar polynomial differential systems \begin{align*}\dot x=ax-y P_n(x,y),\\ \dot y=x ay Q_n(x,y), \end{align*} where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n\geq2$, and $a\in\mathbb R$. Consider the functions \begin{align*} &\varphi(\theta)=P_n(\cos\theta,\sin\theta)\cos\theta Q_n(\cos\theta,\sin\theta)\sin\theta,\\ &\psi(\theta)=Q_n(\cos\theta,\sin\theta)\cos\theta-P_n(\cos\theta,\sin\theta)\sin\theta,\\ &\omega_1(\theta)=a\psi(\theta)-\varphi(\theta),\\ &\omega_2(\theta)=(n-1)\big(2a\psi(\theta)-\varphi(\theta)\big) \psi'(\theta). \end{align*}First we prove that these differential systems have at most $1$ limit cycle if there exists a linear combination of $\omega_1$ and
|
|
$\omega_2$ with definite sign. This result improves previous knwon results. Furthermore, if $\omega_1(\nu_1a\psi-\nu_2\varphi)\leq0$ for some $\nu_1,\nu_2\geq0$, we provide necessary and sufficient conditions for the non-existence, and the existence and uniqueness of the limit cycles of these differential systems. When one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin.
|
|
file: HuaLiaLli2017.preprint.pdf-b7adbcb909f90cba27bdf6e7ce2b95b3.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4374
|
|
bibtexid: CheLliTan2018
|
|
title: Global dynamics of a {S}{D} oscillator
|
|
author: Chen, Hebai
|
|
author: Llibre, Jaume
|
|
author: Tang, Yilei
|
|
journal: Nonlinear Dynamics
|
|
issn: 0924-090X
|
|
year: 2018
|
|
volume: 91
|
|
number: 3
|
|
startpage: 1755
|
|
endpage: 1777
|
|
doi: 10.1007/s11071-017-3979-y
|
|
keywords: averaging method
|
|
keywords: Bogdanov-Takens bifurcation
|
|
keywords: homoclinic loop
|
|
keywords: Hopf bifurcation
|
|
keywords: limit cycles
|
|
keywords: SD oscillator
|
|
abstract: In this paper we derive the global bifurcation diagrams of a SD oscillator which exhibits both smooth and discontinuous dynamics depending on the value of a parameter $a$. We research all possible bifurcations of this system, including Pitchfork bifurcation, degenerate Hopf bifurcation, Homoclinic bifurcation, Double limit cycle bifurcation, Bautin bifurcation and Bogdanov-Takens bifurcation. Besides we prove that the system has at most five limit cycles. At last, we give all numerical phase portraits to illustrate our results.
|
|
file: CheLliTan2016.Preprint.pdf-f0641ee91ac7ae20adb016855d22a96e.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4391
|
|
bibtexid: LliMarVid2018
|
|
title: Linear type centers of polynomial {H}amiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis
|
|
author: Llibre, Jaume
|
|
author: Martinez Mancilla, Yohanna Paulina
|
|
author: Vidal, Claudio
|
|
journal: Discrete and Continuous Dynamical Systems. Series B
|
|
issn: 1531-3492
|
|
year: 2018
|
|
volume: 23
|
|
number: 2
|
|
startpage: 887
|
|
endpage: 912
|
|
doi: 10.3934/dcdsb.2018047
|
|
keywords: Hamiltonian systems
|
|
keywords: linear type centers
|
|
keywords: phase portraits
|
|
keywords: Polynomial vector fields
|
|
keywords: quartic polynomial
|
|
abstract: We provide normal forms and the phase portraits in the Poincar\'{e} disk for all the linear type centers of polynomial Hamiltonian systems with nonlinearities of degree $4$ symmetric with respect to the $y$-axis.
|
|
file: LliMarVid2016.Preprint.pdf-c66f6fd98fedcf14a159a6dfa23e364c.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4405
|
|
bibtexid: LliYu2018a
|
|
title: {N}o periodic orbits in the {B}ianchi models {B}
|
|
author: Llibre, Jaume
|
|
author: Yu, Jiang
|
|
journal: Journal of Geometry and Physics
|
|
issn: 0393-0440
|
|
year: 2018
|
|
volume: 128
|
|
startpage: 32
|
|
endpage: 37
|
|
doi: 10.1016/j.geomphys.2018.01.026
|
|
keywords: Bianchi cosmological models
|
|
keywords: periodic orbit
|
|
abstract: In this paper we prove that the Bianchi models B have no periodic solutions.
|
|
file: LliYu2016.preprint.pdf-f4b2b7daea019dfc6e9e88c69e26e2ff.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4488
|
|
bibtexid: LliVal2018a
|
|
title: Global phase portraits for the {A}bel quadratic polynomial differential equations of second kind with {Z}_2-symmetries
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Canadian Mathematical Bulletin
|
|
issn: 1496-4287
|
|
year: 2018
|
|
volume: 61
|
|
number: 1
|
|
startpage: 149
|
|
endpage: 165
|
|
doi: 10.4153/CMB-2017-026-6
|
|
keywords: Abel polynomial differential systems of the second kind
|
|
keywords: Phase portrait
|
|
keywords: vector fields
|
|
abstract: We provide normal forms and the global phase portraits in the Poincar\'e disk for all Abel quadratic polynomial differential equations of the second kind with $\Z_2$-symmetries.
|
|
file: LliVal2016g.preprint.pdf-1ef52b98d5c824919a5e18624dd0afe9.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4514
|
|
bibtexid: CorCorRob2018
|
|
title: A Four-Body Convex Central Configuration with Perpendicular Diagonals Is Necessarily a Kite
|
|
author: Corbera, Montserrat
|
|
author: Cors, Josep Maria
|
|
author: Roberts, Gareth E.
|
|
journal: Qualitative Theory of Dynamical Systems
|
|
issn: 1575-5460
|
|
year: 2018
|
|
volume: 17
|
|
number: 2
|
|
startpage: 367
|
|
endpage: 374
|
|
doi: 10.1007/s12346-017-0238-z
|
|
keywords: Central configuration
|
|
keywords: n-body problem
|
|
keywords: n-vortex problem
|
|
abstract: We prove that any four-body convex central configuration with perpendicular diagonals must be a kite configuration. The result extends to general power-law potential functions, including the planar four-vortex problem.
|
|
file: CorCorRob2016.preprint.pdf-c3d73c0db6179dab381eb59d8a5ec1ca.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4518
|
|
bibtexid: LliVal2018d
|
|
title: Algebraic limit cycles for quadratic polynomial differential systems
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Discrete and Continuous Dynamical Systems. Series B
|
|
issn: 1531-3492
|
|
year: 2018
|
|
volume: 23
|
|
number: 6
|
|
startpage: 2475
|
|
endpage: 2485
|
|
doi: 10.3934/dcdsb.2018070
|
|
keywords: Algebraic limit cycles
|
|
keywords: quadratic polynomial differential system
|
|
keywords: quadratic polynomial vector field
|
|
abstract: We prove that for a quadratic polynomial differential system having three pairs of diametrally opposite equilibrium points at infinity that are positively rationally independent, has at most one algebraic limit cycle. Our result provides a partial positive answer to the following conjecture: Quadratic polynomial differential systems have at most one algebraic limit cycle.
|
|
file: LliVal2017b.preprint.pdf-d2643a254bc8f0c9ff5e2b3a9b63f73a.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4521
|
|
bibtexid: CimGasMan2018a
|
|
title: Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points
|
|
author: Cima, Anna
|
|
author: Gasull, Armengol
|
|
author: Mañosa, Víctor
|
|
journal: Discrete and Continuous Dynamical Systems. Series A
|
|
issn: 1078-0947
|
|
year: 2018
|
|
volume: 38
|
|
number: 2
|
|
startpage: 889
|
|
endpage: 904
|
|
doi: 10.3934/dcds.2018038
|
|
keywords: local and global asymptotic stability
|
|
keywords: non-hyperbolic points
|
|
keywords: Parrondo's dynamic paradox
|
|
keywords: Periodic discrete dynamical systems
|
|
abstract: We show that for periodic non-autonomous discrete dynamical systems, even when a common fixed point for each of the autonomous associated dynamical systems is repeller, this fixed point can became a local attractor for the whole system, giving rise to a Parrondo's dynamic type paradox.
|
|
file: CimGasMan2017.preprint.pdf-01bc723879bbd5dee15cad57379ba303.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4398
|
|
bibtexid: ManRojVil2018
|
|
title: Analytic tools to bound the criticality at the outer boundary of the period annulus
|
|
author: Mañosas, Francesc
|
|
author: Rojas, David
|
|
author: Villadelprat, Jordi
|
|
journal: Journal of Dynamics and Differential Equations
|
|
issn: 1572-9222
|
|
year: 2018
|
|
volume: 30
|
|
number: 3
|
|
startpage: 883
|
|
endpage: 909
|
|
doi: 10.1007/s10884-016-9559-x
|
|
keywords: Bifurcation
|
|
keywords: Center
|
|
keywords: Chebyshev system
|
|
keywords: critical periodic orbit
|
|
keywords: criticality
|
|
keywords: Period function
|
|
abstract: In this paper we consider planar potential differential systems and we study the bifurcation of critical periodic orbits from the outer boundary of the period annulus of a center. In the literature the usual approach to tackle this problem is to obtain a uniform asymptotic expansion of the period function near the outer boundary. The novelty in the present paper is that we directly embed the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary. We obtain in this way explicit sufficient conditions in order that at most n 0 critical periodic orbits bifurcate from the outer boundary. These theoretical results are then applied to study the bifurcation diagram of the period function of the family x ̈ = xp − xq , p, q ∈ R with p > q.
|
|
file: ManRojVil2016b.preprint.pdf-537061e47e47c69a8631ec0524e9dc83.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4424
|
|
bibtexid: LliLopMor2018
|
|
title: Limit cycles bifurcating from the periodic orbits of the weight-homogeneous polynomial centers of weight-degree 3
|
|
author: Llibre, Jaume
|
|
author: Lopes, Bruno D.
|
|
author: de Moraes, Jaime R.
|
|
journal: Electronic Journal of Differential Equations
|
|
issn: 1072-6691
|
|
year: 2018
|
|
number: 118
|
|
startpage: 1
|
|
endpage: 14
|
|
keywords: averaging method
|
|
keywords: limit cycles
|
|
keywords: Polinomial vector field
|
|
keywords: weight-homogeneous differential system
|
|
abstract: In this paper we obtain two explicit polynomials, whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of polynomial differential centers of order 5, when this family is perturbed inside the class of all polynomial differential systems of order 5, whose average function of first order is not zero. Then the maximum number of limit cycles that bifurcate from these periodic orbits is 6 and it is reached. The family of centers studied completes the study about the limit cycles which can bifurcate from the periodic orbits of all centers of the weight--homogeneous polynomial differential systems of weight--degree 3, when we perturb them inside the class of all polynomial differential systems having the same degree, and whose
|
|
average function of first order is not zero.
|
|
file: LliLopMor2016a.preprint.pdf-a340c8f1906a1d6f91af08a91e570f36.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4434
|
|
bibtexid: GarLliMaz2018
|
|
title: On the multiple zeros of a real analytic function with applications to the averaging theory of differential equations
|
|
author: García, Isaac A.
|
|
author: Llibre, Jaume
|
|
author: Maza, Susanna
|
|
journal: Nonlinearity
|
|
issn: 1361-6544
|
|
year: 2018
|
|
volume: 31
|
|
number: 6
|
|
startpage: 2666
|
|
endpage: 2688
|
|
doi: 10.1088/1361-6544/aab592
|
|
keywords: Averaging theory
|
|
keywords: periodic orbit
|
|
keywords: Poincaré map
|
|
abstract: In this work we improve the classical averaging theory applied to $\lambda$-families of analytic $T$-periodic ordinary differential equations in standard form defined on $\mathbb{R}$. First we characterize the set of points $z_0$ in the phase space and the parameters $\lambda$ where $T$-periodic solutions can be produced when we vary a small parameter $\varepsilon$. Second we expand the displacement map in powers of the parameter $\varepsilon$ whose coefficients are the averaged functions. The main contribution consists in analyzing the role that have the multiple zeros $z_0 \in\mathbb{R}$ of the first non-zero averaged function. The outcome is that these multiple zeros can be of two different classes depending on whether the points $(z_0, \lambda)$ belong or not to the analytic set defined by the real variety associated to the ideal generated by the averaged functions in the Noetheriang ring of all the real analytic functions at $(z_0, \lambda)$. Next we are able to bound the maximum number of branches of isolated $T$-periodic solutions that can bifurcate from each multiple zero $z_0$. Sometimes these bounds depend on the cardinalities of minimal bases of the former ideal. Several examples illustrate our results.
|
|
file: GarLliMaz2016a.Preprint.pdf-06d68be8d027dbf3034f6a8c209d1cd2.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4522
|
|
bibtexid: LliOliVal2018
|
|
title: Phase portraits for some symmetric cubic {R}iccati polynomial differential equations
|
|
author: Llibre, Jaume
|
|
author: Oliveira, Regilene D. S.
|
|
author: Valls, Clàudia
|
|
journal: Topology and its Applications
|
|
issn: 0166-8641
|
|
year: 2018
|
|
volume: 234
|
|
startpage: 220
|
|
endpage: 237
|
|
doi: 10.1016/j.topol.2017.11.023
|
|
keywords: equivariance
|
|
keywords: Phase portrait
|
|
keywords: reversibility
|
|
keywords: Riccati polynomial differential systems
|
|
keywords: vector fields
|
|
abstract: We classify the topological phase portraits in the Poincaré disc of two classes of symmetric Riccati cubic polynomial differential systems.
|
|
file: LliOliVal2017.preprint.pdf-76160d0a1c0770ad51de4c67fe9ac1f9.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4523
|
|
bibtexid: CorLliVal2018
|
|
title: Periodic orbits of perturbed non-axially symmetric potentials in 1:1:1 and 1:1:2 resonances
|
|
author: Corbera, Montserrat
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Discrete and Continuous Dynamical Systems. Series B
|
|
issn: 1531-3492
|
|
year: 2018
|
|
volume: 23
|
|
number: 6
|
|
startpage: 2299
|
|
endpage: 2337
|
|
doi: 10.3934/dcdsb.2018101
|
|
keywords: Averaging theory
|
|
keywords: galactic potential
|
|
keywords: periodic solution
|
|
abstract: We analytically study the Hamiltonian system in $\mathbb{R}^6$ with Hamiltonian $$ H= \frac12 (p_x^2 p_y^2 p_z^2) \frac{1}{2} (\omega_1^2 x^2 \omega_2^2 y^2 \omega_3^2 z^2)
|
|
\varepsilon(a z^3 z (b x^2 c y^2)), $$ being $a,b,c\in\mathbb{R}$ with $c\ne 0$, $\varepsilon$ a small parameter, and $\omega_1$, $\omega_2$ and $\omega_3$ the unperturbed frequencies of the oscillations along the $x$, $y$ and $z$ axis, respectively. For $|\varepsilon|>0$ small, using averaging theory of first and second order we find periodic orbits in every positive energy level of $H$ whose frequencies are $\omega_1=\omega_2=\omega_3/2$ and $\omega_1=\omega_2=\omega_3$, respectively (the number of such periodic orbits depends on the values of the parameters $a,b,c$). We also provide the shape of the periodic orbits and their linear stability.
|
|
file: CorLliVal2017.preprint.pdf-72721c80094c9a930a5d1cf2c784ca57.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4538
|
|
bibtexid: LliSir2018
|
|
title: On {L}efschetz periodic point free self-maps
|
|
author: Llibre, Jaume
|
|
author: Sirvent, Víctor F.
|
|
journal: Journal of Fixed Point Theory and Applications
|
|
issn: 1661-7746
|
|
year: 2018
|
|
volume: 30
|
|
number: 38
|
|
doi: 10.1007/s11784-018-0498-5
|
|
keywords: Lefschetz numbers
|
|
keywords: Lefschetz zeta function
|
|
keywords: periodic point
|
|
keywords: product of spheres
|
|
keywords: wedge sum of spheres
|
|
abstract: We study the periodic point free maps on connected retract of a finite simplicial complex using the Lefschetz numbers. We put special emphasis in the self-maps on the product of spheres and of the wedge sums of spheres.
|
|
file: LliSir2017.preprint.pdf-d9f61976f3314c51925e95dc81915994.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4542
|
|
bibtexid: LliTon2018
|
|
title: The symmetric periodic orbits for the classical helium atom
|
|
author: Llibre, Jaume
|
|
author: Tonon, Durval J.
|
|
journal: Letters in Mathematical Physics
|
|
issn: 1573-0530
|
|
year: 2018
|
|
volume: 108
|
|
number: 8
|
|
startpage: 1851
|
|
endpage: 1871
|
|
doi: 10.1007/s11005-018-1056-1
|
|
keywords: helium atom
|
|
keywords: Poincar\'{e} continuation method
|
|
keywords: symmetric periodic orbits
|
|
abstract: We analyse the existence of periodic symmetric orbits of the classical helium atom. The results obtained shows that there exists six families of periodic orbits that can be prolonged from a continuum of periodic symmetric orbits. The main technique applied in this study is the continuation method of Poincaré.
|
|
file: LliTon2017a.preprint.pdf-cef2474eacf592076f36964061ce9eeb.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4548
|
|
bibtexid: CanLli2018
|
|
title: Stability of periodic orbits in the averaging theory: Applications to {L}orenz and {T}homas' differential systems
|
|
author: Cândido, Murilo R.
|
|
author: Llibre, Jaume
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
issn: 0218-1274
|
|
year: 2018
|
|
volume: 28
|
|
number: 3
|
|
startpage: 1830007
|
|
doi: 10.1142/S0218127418300070
|
|
keywords: Averaging theory
|
|
keywords: Circulant systems
|
|
keywords: Lorenz
|
|
keywords: stability of periodic orbits
|
|
keywords: Thomas
|
|
abstract: We study the kind of stability of the periodic orbits provided by higher order averaging theory. We apply these results for determining the $k-$hyperbolicity of some periodic orbits of the Lorenz and Thoma's differential system.
|
|
file: CanLli2017.preprint.pdf-8bf2a0b70594b29715c1a0aefc96100b.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4528
|
|
bibtexid: BarLliVal2018
|
|
title: Limit cycles bifurcating from a zero-{H}opf singularity in arbitrary dimension
|
|
author: Barreira, Luis
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Nonlinear Dynamics
|
|
issn: 0924-090X
|
|
year: 2018
|
|
volume: 92
|
|
number: 3
|
|
startpage: 1159
|
|
endpage: 1166
|
|
doi: 10.1007/s11071-018-4115-3
|
|
abstract: We study the limit cycles which can bifurcate from a zero--Hopf singularity of a $C^{m 1}$ differential system in $\R^n$, i.e. from a singularity with eigenvalues $\pm b i$ and $n-2$ zeros for $n\ge 3$. If this singularity is at the origin of coordinates and the Taylor expansion of the differential system at the origin without taking into account the linear terms starts with terms of order $m$, from the origin it can bifurcate $s$ limit cycles with $s\in \{ 0,1,\ldots, 2^{n-3}\}$ if $m=2$ (see \cite{LZ}), with $s\in \{ 0,1,\ldots, 3^{n-2}\}$ if $m=3$, with $s\le 6^{n-2}$ if $m=4$, and with $s\le 4\cdot 5^{n-2}$ if $m=5$. Moreover, $s\in \{0,1,2\}$ if $m=4$ and $n=3$, and $s\in \{0,1,2,3,4,5\}$ if $m=5$ and $n=3$. Note that the maximum number of limit cycles bifurcating from this zero--Hopf singularity grows up exponentially with the dimension for $m=2,3$.
|
|
file: BarLliVal2017.preprint.pdf-c5ea3bfa2c728380685d7555c892fdf8.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4531
|
|
bibtexid: LliMur2018
|
|
title: Darboux theory of integrability for real polynomial vector fields on $\mathbb {S}^n$
|
|
author: Llibre, Jaume
|
|
author: Murza, Adrian
|
|
journal: Dynamical Systems. An International Journal
|
|
issn: 1468-9367
|
|
year: 2018
|
|
startpage: 1
|
|
endpage: 14
|
|
doi: 10.1080/14689367.2017.1420141
|
|
keywords: Darboux integrability theory
|
|
keywords: invariant meridian
|
|
keywords: invariant parallel
|
|
keywords: n--dimensional spheres
|
|
abstract: This is a survey on the Darboux theory of integrability for
|
|
polynomial vector fields, first in $\R^n$ and second in the
|
|
$n$-dimensional sphere $\sss^n$. We also provide new results about
|
|
the maximum number of invariant parallels and meridians that a polynomial
|
|
vector field $\X$ on $\sss^n$ can have in function of its degree.
|
|
These results in some sense extend the known result on the maximum
|
|
number of hyperplanes that a polynomial vector field $\Y$ in $\R^n$
|
|
can have in function of the degree of $\Y$.
|
|
file: LliMur2017.preprint.pdf-e1276fd6df078deb983029fef24713b1.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4547
|
|
bibtexid: RojVil2018
|
|
title: A criticality result for polycycles in a family of quadratic reversible centers
|
|
author: Rojas, David
|
|
author: Villadelprat, Jordi
|
|
journal: Journal of Differential Equations
|
|
issn: 0022-0396
|
|
year: 2018
|
|
volume: 264
|
|
number: 11
|
|
startpage: 6585
|
|
endpage: 6602
|
|
doi: 10.1016/j.jde.2018.01.042
|
|
keywords: Bifurcation
|
|
keywords: Center
|
|
keywords: critical periodic orbit
|
|
keywords: criticality
|
|
keywords: Period function
|
|
abstract: We consider the family of dehomogenized LoudÕs centers $X_{\mu}=y(x-1)\partial_x (x Dx^2 Fy^2)\partial_y,$
|
|
where $\mu=(D,F)\in\R^2,$ and we study the number of critical periodic orbits that emerge or dissapear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family $\{X_{\mu},\mu\in\R^2\}$ distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set $\Gamma_{B}$ of codimension 1 in $\R^2$. In the present paper we succeed in proving that a subset of $\Gamma_{B}$ has criticality equal to one.
|
|
file: RojVil2017.preprint.pdf-6e30de8beeaf41bca06cd4e32f3d5024.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4559
|
|
bibtexid: LliVal2018g
|
|
title: Algebraic limit cycles on quadratic polynomial differential systems
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Proceedings of the Edinburgh Mathematical Society
|
|
issn: 0013-0915
|
|
year: 2018
|
|
volume: 61
|
|
number: 2
|
|
startpage: 499
|
|
endpage: 512
|
|
doi: 10.1017/S0013091517000244
|
|
keywords: Algebraic limit cycles
|
|
keywords: quadratic polynomial differential system
|
|
keywords: quadratic polynomial vector field
|
|
abstract: Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared: Quadratic polynomial differential systems have at most one algebraic limit cycle.
|
|
We prove that for a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrally opposite singular point at infinity, has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.
|
|
file: LliVal2017g.preprint.pdf-96e3f03c593c599ca0aab7a6b073e3f4.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4574
|
|
bibtexid: BuzLimTor2018
|
|
title: Limit cycles via higher order perturbations for some piecewise differential systems
|
|
author: Buzzi, Claudio A.
|
|
author: Lima, Mauricio Firmino Silva
|
|
author: Torregrosa, Joan
|
|
journal: Physica D. Nonlinear Phenomena
|
|
issn: 0167-2789
|
|
year: 2018
|
|
volume: 371
|
|
startpage: 28
|
|
endpage: 47
|
|
doi: 10.1016/j.physd.2018.01.007
|
|
keywords: Liénard piecewise differential system
|
|
keywords: limit cycle in Melnikov higher order perturbation
|
|
keywords: non-smooth differential system
|
|
abstract: A classical perturbation problem is the polynomial perturbation of the harmonic oscillator, $(x',y')=(-y \varepsilon f(x,y,\varepsilon),x \varepsilon g(x,y,\varepsilon)).$ In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line. We prove that, for polynomial perturbations of degree $n,$ no more than $Nn-1$ limit cycles appear up to a study of order $N$. We also show that this upper bound is reached for orders one and two. Moreover, we study this problem in some classes of piecewise Li\'enard differential systems. When we restrict the analysis to some special class this upper bound never is attained and we show which is this upper bound for higher order perturbation in $\varepsilon$. The Poincar\'e--Pontryagin--Melnikov theory is the main technique used to prove all the results.
|
|
file: BuzLimTor2017.preprint.pdf-74f9fdbcd1b3892e5a79ac4bc09d013d.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4576
|
|
bibtexid: FerValWiu2018
|
|
title: On {D}arboux integrability of {E}delstein's reaction system in $\mathbb {R}^3$
|
|
author: Ferragut, Antoni
|
|
author: Valls, Clàudia
|
|
author: Wiuf, Carsten
|
|
journal: Chaos, Solitons and Fractals
|
|
issn: 0960-0779
|
|
year: 2018
|
|
volume: 108
|
|
startpage: 129
|
|
endpage: 135
|
|
doi: 10.1016/j.chaos.2018.01.029
|
|
keywords: deficiency theorem
|
|
keywords: Exponential factor
|
|
keywords: First integral
|
|
keywords: multi-stationarity
|
|
keywords: polynomial system
|
|
keywords: Reaction network
|
|
abstract: We consider Edelstein's dynamical system of three reversible reactions in $\mathbb R^3$ and show that it is not Darboux integrable. To do so we characterize its polynomial first integrals, Darboux polynomials and exponential factors.
|
|
file: FerValWiu2017.preprint.pdf-df8d0758e0803810ffcaab90a025a79d.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4578
|
|
bibtexid: FerGarLliMel2018
|
|
title: New central configurations of the $(n 1)$--body problem
|
|
author: Fernandes, Antonio Carlos
|
|
author: Garcia, Braulio Augusto
|
|
author: Llibre, Jaume
|
|
author: Mello, Luis Fernando
|
|
journal: Journal of Geometry and Physics
|
|
issn: 0393-0440
|
|
year: 2018
|
|
volume: 124
|
|
startpage: 199
|
|
endpage: 207
|
|
doi: 10.1016/j.geomphys.2017.11.003
|
|
keywords: $(n 1)$--body problem
|
|
keywords: Celestial Mechanics
|
|
keywords: Central configuration
|
|
abstract: In this article we study central configurations of the $(n 1)$--body problem. For the planar $(n 1)$--body problem we study central configurations performed by $n \geq 2$ bodies with equal masses at the vertices of a regular $n$--gon and one body with null mass. We also study spatial central configurations considering $n$ bodies with equal masses at the vertices of a regular polyhedron and one body with null mass.
|
|
file: FerGarLliMel2017.pdf-ba095af5a51f188235ebbb74fd9f089c.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4590
|
|
bibtexid: DiaLliVal2018
|
|
title: Polynomial {H}amiltonian systems of degree 3 with symmetric nilpotent centers
|
|
author: Dias, Fabio Scalco
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Mathematics and Computers in Simulation
|
|
issn: 0378-4754
|
|
year: 2018
|
|
volume: 144
|
|
startpage: 60
|
|
endpage: 77
|
|
doi: 10.1016/j.matcom.2017.06.002
|
|
keywords: Nilpotent center
|
|
keywords: Phase portrait
|
|
keywords: Poincaré compactification
|
|
keywords: polynomial Hamiltonian systems
|
|
abstract: We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the $x$-axis having a nilpotent center at the origin.
|
|
file: DiaLliVal2018.Preprint.pdf-2863f40a6c46a690f34d3d3bb2484327.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4591
|
|
bibtexid: ArtLliVal2018
|
|
title: Dynamics of the {H}iggins-{S}elkov and {S}elkov systems
|
|
author: Artés, Joan Carles
|
|
author: Llibre, Jaume
|
|
author: Valls, Clàudia
|
|
journal: Chaos, Solitons and Fractals
|
|
issn: 0960-0779
|
|
year: 2018
|
|
volume: 114
|
|
startpage: 145
|
|
endpage: 150
|
|
doi: 10.1016/j.chaos.2018.07.007
|
|
keywords: Higgins-Selkov system
|
|
keywords: Phase portrait
|
|
keywords: Poincaré compactification
|
|
keywords: Selkov system
|
|
abstract: We describe the global dynamics in the Poincar\'e disc of the Higgins--Selkov model \begin{equation*} x'= k_0 -k_1 x y^2, \quad y'= -k_2 y k_1 x y^2, \end{equation*} where $k_0,k_1,k_2$ are positive parameters, and of the Selkov model
|
|
\begin{equation*} x'= - x a y x^2 y, \quad y'= b - a y - x^2 y, \end{equation*} where $a,b$ are positive parameters.
|
|
file: ArtLliVal2017.preprint.pdf-a25fc1e8a7b05d28abda8c807be68ed8.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4575
|
|
bibtexid: CruTor2018
|
|
title: Simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli
|
|
author: da Cruz, Leonardo Pereira Costa
|
|
author: Torregrosa, Joan
|
|
journal: Journal of Mathematical Analysis and Applications
|
|
issn: 0022-247X
|
|
year: 2018
|
|
volume: 461
|
|
startpage: 248
|
|
endpage: 272
|
|
doi: 10.1016/j.jmaa.2017.12.072
|
|
keywords: limit cycles
|
|
keywords: Piecewise vector field
|
|
keywords: Simultaneous bifurcation
|
|
keywords: Zeros of Abelian integrals
|
|
abstract: We study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations. The cubic planar system $(x',y')=-y((x-1)^2 y^2),x((x-1)^2 y^2)$ has simultaneously a center at the origin and at infinity. We study, up to first order averaging analysis, the bifurcation of periodic orbits from the two period annuli first separately and second simultaneously. This problem is an generalization of \cite{PerTor2014} to the piecewise systems class. When the polynomial perturbation has degree $n$, we prove that the inner and outer Abelian integrals are rational functions and we provide an upper bound for the number of zeros. When the perturbation is cubic, the same degree than the unperturbed vector field, the maximum number of limit cycles, up to first order perturbation, from the inner and outer annuli is 9 and 8, respectively. But, when the simultaneous bifurcation problem is considered, 12 limit cycles exist. These limit cycles appear in three type of configurations: (9,3), (6,6) and (4,8). In the non-piecewise scenario only 5 limit cycles were found.
|
|
file: CruTor2017.Preprint.pdf-81ed6836b6e822e28ce0bc04148384ae.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4589
|
|
bibtexid: LliTei2018
|
|
title: Piecewise linear differential systems with only centers can create limit cycles?
|
|
author: Llibre, Jaume
|
|
author: Teixeira, Marco Antonio
|
|
journal: Nonlinear Dynamics
|
|
issn: 0924-090X
|
|
year: 2018
|
|
volume: 91
|
|
number: 1
|
|
startpage: 249
|
|
endpage: 255
|
|
doi: 10.1007/s11071-017-3866-6
|
|
keywords: continuous piecewise linear differential system
|
|
keywords: discontinuous piecewise differential systems
|
|
keywords: First integral
|
|
keywords: limit cycles
|
|
keywords: Linear centers
|
|
abstract: In this article, we study the continuous and discontinuous planar piecewise differential systems formed only by linear centers separated by one or two parallel straight lines. When these piecewise differential systems are continuous, they have no limit cycles. Also if they are discontinuous separated by a unique straight line, they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle.
|
|
file: LliTei2017b.preprint.pdf-0ad97dd83da0ea62e1fd6f35be61c0c0.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4604
|
|
bibtexid: BuzGasTor2018
|
|
title: Algebraic limit cycles in piecewise linear differential systems
|
|
author: Buzzi, Claudio A.
|
|
author: Gasull, Armengol
|
|
author: Torregrosa, Joan
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
issn: 0218-1274
|
|
year: 2018
|
|
volume: 28
|
|
number: 3
|
|
startpage: 1850039
|
|
doi: 10.1142/S0218127418500396
|
|
keywords: Algebraic limit cycles
|
|
keywords: hyperbolic and double limit cycle
|
|
keywords: non-smooth differential system
|
|
keywords: Piecewise linear systems
|
|
keywords: saddle-node bifurcation
|
|
abstract: This paper is devoted to study the algebraic limit cycles of planar piecewise linear differential systems. In particular we present examples exhibiting two explicit hyperbolic algebraic limit cycles, as well as some 1-parameter families with a saddle-node bifurcation of algebraic limit cycles. We also show that all degrees for algebraic limit cycles are allowed.
|
|
file: BuzGasTor2018.Preprint.pdf-76638afc6c3be306e1c3a48442386e47.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4610
|
|
bibtexid: HanLliYan2018
|
|
title: On uniqueness of limit cycles in general {B}ogdanov-{T}akens bifurcation
|
|
author: Han, Maoan
|
|
author: Llibre, Jaume
|
|
author: Yang, Junmin
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
issn: 0218-1274
|
|
year: 2018
|
|
volume: 28
|
|
number: 9
|
|
startpage: 1850115
|
|
doi: 10.1142/S0218127418501158
|
|
keywords: Bodganov-Takens bifurcation
|
|
abstract: In this paper we present a complete study to the well-known Bogdanov-Takens bifurcation and give a rigorous proof for the uniqueness of limit cycles.
|
|
file: HanLliYan2017.preprint.pdf-37647794c2cee7c4b07ed2db707d63a5.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4637
|
|
bibtexid: OllRodSol2018
|
|
title: Ejection-collision orbits in the {R}{T}{B}{P}
|
|
author: Ollé, Mercè
|
|
author: Rodríguez, Òscar
|
|
author: Soler, Jaume
|
|
journal: Communications in Nonlinear Science and Numerical Simulation
|
|
issn: 1007-5704
|
|
year: 2018
|
|
volume: 55
|
|
startpage: 298
|
|
endpage: 315
|
|
doi: 10.1016/j.cnsns.2017.07.013
|
|
keywords: bifurcations
|
|
keywords: Ejection-collision orbits
|
|
keywords: invariant manifolds
|
|
keywords: regularization
|
|
abstract: In this paper we analyse the ejection-collision (EC) orbits of the planar restricted three body problem. Being μ∈(0, 0.5] the mass parameter, and taking the big (small) primary with mass 1 − μ (μ), an EC orbit will be an orbit that ejects from the big primary, does an excursion and collides with it. As it is well known, for any value of the mass parameter μ∈(0, 0.5] and sufficiently restricted Hill regions (that is, for big enough values of the Jacobi constant C), there are exactly four EC orbits. We check their existence and extend numerically these four orbits for μ∈(0, 0.5] and for smaller values of the Jacobi constant. We introduce the concept of n-ejection-collision orbits (n-EC orbits) and we explore them numerically for μ∈(0, 0.5] and values of the Jacobi constant such that the Hill bounded possible region of motion contains the big primary and does not contain the small one. We study the cases 1≤n≤10 and we analyse the continuation of families of such n-EC orbits, varying the energy, as well as the bifurcations that appear.
|
|
file: OllRodSol2018.preprint.pdf-18d6c8e4be63ed1164acbd5bd48f896a.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4615
|
|
bibtexid: CenLliZha2018
|
|
title: Periodic solutions and their stability of some higher-order positively homogenous differential equations
|
|
author: Cen, Xiuli
|
|
author: Llibre, Jaume
|
|
author: Zhang, Meirong
|
|
journal: Chaos, Solitons and Fractals
|
|
issn: 0960-0779
|
|
year: 2018
|
|
volume: 106
|
|
startpage: 285
|
|
endpage: 288
|
|
doi: 10.1016/j.chaos.2017.11.032
|
|
keywords: $m$-Order differential equation
|
|
keywords: Averaging theory
|
|
keywords: periodic solution
|
|
keywords: Stability
|
|
abstract: In the present paper we study periodic solutions and their stability of the $m$-order differential equations of the form $$ x^{(m)} f_n(x) = \mu h(t), $$ where the integers $m, n\geq2$, $f_n(x)= \da x^n$ or $\da |x|^n$ with $\da=\pm 1$, and $h(t)$ is a continuous $T$-periodic function of non-zero average, and $\mu$ is a positive small parameter. By using the averaging theory, we will give the existence of $T$-periodic solutions. Moreover, the instability and the linear stability of these periodic solutions will be obtained.
|
|
file: CenLliZha2018.preprint.pdf-c3a3054b33d1959025b4998e7a6bfa4c.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4631
|
|
bibtexid: OrtRoj2018
|
|
title: A proof of {B}ertrand’s theorem using the theory of isochronous potentials
|
|
author: Ortega, Rafael
|
|
author: Rojas, David
|
|
journal: Journal of Dynamics and Differential Equations
|
|
issn: 1572-9222
|
|
year: 2018
|
|
doi: 10.1007/s10884-018-9676-9
|
|
keywords: Bertrand’s theorem
|
|
keywords: Isochronicity
|
|
keywords: potential center
|
|
abstract: We give an alternative proof for the celebrated Bertrand’s theorem as a corollary of the isochronicity of a certain family of centers.
|
|
file: OrtRoj2018.preprint.pdf-fbb8d5c5a50d74f44e6d6b0c33d26bc6.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4633
|
|
bibtexid: GasLloMan2018
|
|
title: Periodic points of a {L}anden transformation
|
|
author: Gasull, Armengol
|
|
author: Llorens, Mireia
|
|
author: Mañosa, Víctor
|
|
journal: Communications in Nonlinear Science and Numerical Simulation
|
|
issn: 1007-5704
|
|
year: 2018
|
|
volume: 64
|
|
startpage: 232
|
|
endpage: 245
|
|
doi: 10.1016/j.cnsns.2018.04.020
|
|
keywords: Landen transformation
|
|
keywords: periodic points
|
|
keywords: Poincaré-Miranda theorem
|
|
abstract: We prove the existence of 3-periodic orbits in a dynamical system associated to a Landen transformation previously studied by Boros, Chamberland and Moll, disproving a conjecture on the dynamics of this planar map introduced by the latter author. To this end we present a systematic methodology to determine and locate analytically isolated periodic points of algebraic maps. This approach can be useful to study other discrete dynamical systems with algebraic nature. Complementary results on the dynamics of the map associated with the Landen transformation are also presented.
|
|
file: GasLloMan2018.preprint.pdf-2db49e7b3e3a5ee24a8af8bbc6a80d8c.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4638
|
|
bibtexid: JuhSal2018
|
|
title: Tuning the overlap and the cross-layer correlations in two-layer networks: application to an {S}{I}{R} model with awareness dissemination
|
|
author: Juher, David
|
|
author: Saldaña, Joan
|
|
journal: Physical Review E
|
|
issn: 2470-0045
|
|
year: 2018
|
|
volume: 97
|
|
number: 3
|
|
startpage: 032303
|
|
doi: 10.1103/PhysRevE.97.032303
|
|
abstract: We study the properties of the potential overlap between two networks $A,B$ sharing the same set of $N$ nodes (a two-layer network) whose respective degree distributions $p_A(k), p_B(k)$ are given. Defining the overlap coefficient $\alpha$ as the Jaccard index, we prove that $\alpha$ is very close to 0 when $A$ and $B$ are random and independently generated. We derive an upper bound $\alpha_M$ for the maximum overlap coefficient permitted in terms of $p_A(k)$, $p_B(k)$ and $N$. Then we present an algorithm based on cross-rewiring of links to obtain a two-layer network with any prescribed $\alpha$ inside the range $(0,\alpha_M)$. A refined version of the algorithm allows us to minimize the cross-layer correlations that unavoidably appear for values of $\alpha$ beyond a critical overlap $\alpha_c<\alpha_M$. Finally, we present a very simple example of an SIR epidemic model with information dissemination and use the algorithms to determine the impact of the overlap on the final outbreak size predicted by the model.
|
|
file: JuhSal2018.preprint.pdf-54df9d825085c21c83f94488b485e532.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4641
|
|
bibtexid: AcoLazMorPan2018
|
|
title: Differential {G}alois theory and non-integrability of planar polynomial vector fields
|
|
author: Acosta-Humánez, Primitivo B.
|
|
author: Lázaro, José Tomás
|
|
author: Morales-Ruiz, Juan J.
|
|
author: Pantazi, Chara
|
|
journal: Journal of Differential Equations
|
|
issn: 0022-0396
|
|
year: 2018
|
|
volume: 264
|
|
number: 12
|
|
startpage: 7183
|
|
endpage: 7212
|
|
doi: 10.1016/j.jde.2018.02.016
|
|
abstract: We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.
|
|
file: AcoLazMorPan2018.preprint.pdf-d62348d62843b9899c8378c02ff51781.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4657
|
|
bibtexid: CanLli2018a
|
|
title: Zero--Hopf bifurcations in 3-dimensional differential systems with no equilibria
|
|
author: Cândido, Murilo R.
|
|
author: Llibre, Jaume
|
|
journal: Mathematics and Computers in Simulation
|
|
issn: 0378-4754
|
|
year: 2018
|
|
volume: 151
|
|
startpage: 54
|
|
endpage: 76
|
|
doi: 10.1016/j.matcom.2018.03.008
|
|
keywords: Averaging theory
|
|
keywords: Periodic solutions
|
|
keywords: quadratic polynomial differential system
|
|
keywords: Zero-Hopf bifurcation.
|
|
abstract: We use averaging theory for studying the Hopf and zero--Hopf bifurcations in some chaotic differential systems. These differential systems have a chaotic attractor and no equilibria. Numerically we show the relation between the existence of the periodic solutions studied in these systems and their chaotic attractors.
|
|
file: CanLli2018a.preprint.pdf-1a8d9e6ed827b7e6dbd54f293d336ae5.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4687
|
|
bibtexid: CanLli2018b
|
|
title: Periodic Orbits Bifurcating from a Nonisolated Zero-{H}opf Equilibrium of three-dimensional Differential Systems Revisited
|
|
author: Cândido, Murilo R.
|
|
author: Llibre, Jaume
|
|
journal: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
|
|
issn: 0218-1274
|
|
year: 2018
|
|
volume: 28
|
|
number: 5
|
|
startpage: 1850058
|
|
doi: 10.1142/S021812741850058X
|
|
keywords: Averaging theory
|
|
keywords: Periodic solutions
|
|
keywords: Polynomial differential systems
|
|
keywords: zero-Hopf bifurcation
|
|
keywords: zero-Hopf equilibrium
|
|
abstract: In this paper, we study the periodic solutions bifurcating from a nonisolated zero–Hopf equilib- rium in a polynomial differential system of degree two in R3. More specifically, we use recent results of averaging theory to improve the conditions for the existence of one or two periodic solutions bifurcating from such a zero–Hopf equilibrium. This new result is applied for studying the periodic solutions of differential systems in R3 having n-scroll chaotic attractors.
|
|
file: CanLli2018b.Preprint.pdf-d157c2527fcdb537c8be5bf26e369233.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4709
|
|
bibtexid: Can2018a
|
|
title: Rational maps with {F}atou components of arbitrarily large connectivity
|
|
author: Canela, Jordi
|
|
journal: Journal of Mathematical Analysis and Applications
|
|
issn: 0022-247X
|
|
year: 2018
|
|
volume: 462
|
|
startpage: 35
|
|
endpage: 56
|
|
doi: 10.1016/j.jmaa.2018.01.061
|
|
abstract: We study the family of singular perturbations of Blaschke products \linebreak $B_{a,\lambda}(z)=z^3\frac{z-a}{1-\overline{a}z} \frac{\lambda}{z^2}$. We analyse how the connectivity of the Fatou components varies as we move continuously the parameter $\lambda$. We prove that all possible escaping configurations of the critical point $c_-(a,\lambda)$ take place within the parameter space.
|
|
In particular, we prove that there are maps $B_{a,\lambda}$ which have Fatou components of arbitrarily large finite connectivity within their dynamical planes.
|
|
file: Can2018.Preprint.pdf-3a7e19115b826ed6c1939af0b18833be.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4710
|
|
bibtexid: CamCanVin2018
|
|
title: Convergence regions for the {C}hebyshev--{H}alley family
|
|
author: Campos, Beatriz
|
|
author: Canela, Jordi
|
|
author: Vindel, Pura
|
|
journal: Communications in Nonlinear Science and Numerical Simulation
|
|
issn: 1007-5704
|
|
year: 2018
|
|
volume: 56
|
|
startpage: 508
|
|
endpage: 525
|
|
doi: 10.1016/j.cnsns.2017.08.024
|
|
abstract: In this paper, we study the dynamical behaviour of the Chebyshev--Halley family applied on a family of degree $n$ polynomials. For $n=2$ we bound the set of parameters for which the iterative methods have convergence regions which do not correspond to the basins of attraction of the roots. We also study the dynamics of indifferent fixed points on the boundary of the regions of parameters with bad behaviour. Finally, we provide a numerical study on the boundedness of the regions of parameters with bad behaviour for the family of degree $n$ polynomials.
|
|
file: CamCanVin2018.Preprint.pdf-8165ca500f1a1584ffad98d32e2e3168.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4619
|
|
bibtexid: GasMan2019
|
|
title: Subseries and signed series
|
|
author: Gasull, Armengol
|
|
author: Mañosas, Francesc
|
|
journal: Communications on Pure & Applied Analysis
|
|
issn: 1553-5258
|
|
year: 2019
|
|
volume: 18
|
|
number: 1
|
|
startpage: 479
|
|
endpage: 492
|
|
doi: 10.3934/cpaa.2019024
|
|
keywords: divergent series
|
|
keywords: Harmonic series
|
|
keywords: random series
|
|
keywords: signed sums
|
|
keywords: subsums
|
|
abstract: For any positive decreasing to zero sequence $a_n$ such
|
|
that $\sum a_n$ diverges we consider the related series $\sum
|
|
k_na_n$ and $\sum j_na_n.$ Here, $k_n$ and $j_n$ are real
|
|
sequences such that $k_n\in\{0,1\}$ and $j_n\in\{-1,1\}.$ We study their convergence and characterize it in terms of the density of 1's in the sequences $k_n$ and $j_n.$ We extend our results to series $\sum m_na_n,$ with $m_n\in\{-1,0,1\}$ and apply them to study some associated random series.
|
|
file: GasMan2017.preprint.pdf-72eac3088022b515405dbc6d691387bd.pdf
|
|
|
|
|
|
type: Article
|
|
aigaionid: 4688
|
|
bibtexid: CheLli2019
|
|
title: Limit cycles of a second-order differential equation
|
|
author: Chen, Ting
|
|
author: Llibre, Jaume
|
|
journal: Applied Mathematics Letters. An International Journal of Rapid Publication
|
|
issn: 0893-9659
|
|
year: 2019
|
|
volume: 88
|
|
startpage: 111
|
|
endpage: 117
|
|
doi: 10.1016/j.aml.2018.08.015
|
|
keywords: Averaging theory
|
|
keywords: limit cycle
|
|
keywords: Mathieu-Duffing type
|
|
abstract: We provide an upper for the maximum number of limit cycles bifurcating from the periodic solutions of $\ddot{x} x=0$, when we perturb this system as follows \ \ddot{x} \varepsilon(1 \cos^m \theta)Q(x,y) x=0, \] where $\varepsilon>0$ is a small parameter, $m$ is an arbitrary non-negative integer, $Q(x,y)$ is a polynomial of degree $n$ and $\theta=\arctan(y/x)$. The main tool used for proving our results is the averaging theory.
|
|
file: CheLli2018b.preprint.pdf-1c960ecb719c4481e53aeb90bcc12eba.pdf
|
|
|
|
|
|
|
|
|
|
|
|
|