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 HindmarshRose 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 zeroHopf or a Hopf bifurcation, and graphically we observed that the periodic orbit starting in those bifurcations is at the beginning of the mentioned perioddoubling 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