Search

We prove that a logarithmic foliation corresponding to a generic line arrangement of $d+1 \geq 3$ lines in the complex plane, with pairwise natural and co-prime residues, is a smooth point of the center set of plane foliations (vector fields) of degree $d$., Comment: The introduction and the last section has been changed. Many typos are corrected

We compute the cyclicity of open period annuli of the following generalized Rayleigh-Li\'enard equation $$\ddot{x}+ax+bx^3-(\lambda_1+\lambda_2 x^2+\lambda_3\dot{x}^2+\lambda_4 x^4+\lambda_5\dot{x}^4+\lambda_6 x^6)\dot{x}=0$$ and the equivalent planar system $X_\lambda$, where the coefficients of the perturbation $\lambda_j$ are independent small parameters and $a, b$ are fixed nonzero constants. Our main tool is the machinery of the so called higher-order Poincar\'e-Pontryagin-Melnikov functions (Melnikov functions $M_n$ for short), combined with the explicit computation of center conditions and the corresponding Bautin ideal. We consider first arbitrary analytic arcs $\varepsilon \to \lambda(\varepsilon)$ and explicitly compute all possible Melnikov functions $M_n$ related to the deformation $X_{ \lambda(\varepsilon)} $. At a second step we obtain exact bounds for the number of the zeros of the Melnikov functions (complete elliptic integrals depending on parameter) in an appropriate complex domain, using a modification of Petrov's method. To deal with the general case of six-parameter deformations $\lambda \to X_\lambda$, we compute first the related Bautin ideal. To do this we carefully study the Melnikov functions up to order three, and then use Nakayama lemma from Algebraic geometry. The principalization of the Bautin ideal (achieved after a blow up) reduces finally the study of general deformations $X_{ \lambda } $ to the study of one-parameter deformations $X_{ \lambda(\varepsilon)} $.

We revisit the bifurcation theory of the Lotka-Volterra quadratic system \begin{eqnarray} X_0 :\left\{\begin{aligned} \dot{x}=& - y -x^2+y^2 ,\\ \dot{y}= &\;\;\;\;x - 2xy \end{aligned} \right. \end{eqnarray} with respect to arbitrary quadratic deformations. The system $X_0$ has a double center, which is moreover isochronous. We show that the deformed system $X_0$ can have at most two limit cycles on the finite plane, with possible distribution $(i,j)$, where $i+j\leq2$. Our approach is based on the study of pairs of bifurcation functions associated to the centers, expressed in terms of iterated path integrals of length two., Comment: 40 pages

We show that the center set of reversible cubic systems, close to the symmetric Hamiltonian system $x'=y, y'= x-x^3$ has two irreducible components of co-dimension two in the parameter space. One of them corresponds to the Hamiltonian stratum, the other to systems which are polynomial pull back of an appropriate linear system

We find an upper bound for the number of limit cycles, bifurcating from the 8-loop of the Duffing oscillator $x"= x-x^{3}$ under the special cubic perturbation $$ x"= x-x^{3}+\lambda_{1}y+\lambda_{2}x^{2}+\lambda_{3}xy+\lambda_{4}x^{2}y . $$, Comment: 4 figures, minor corrections, references added. arXiv admin note: text overlap with arXiv:1306.2340

This is an extended version of two lectures given during the Zagreb Dynamical Systems Workshop, October 22-26, 2018.

We prove that the Zarisky closure of the monodromy group of the polynomial $ x^2 y^2(1-x-y) $ is the symplectic group $Sp(4,\mathbb C)$. This shows that some previous results about this monodromy representation are wrong., Comment: A few typos were corrected

This article introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert's 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor $E$ of the canonical blow-up $B_I{\mathbb C}^n$ of the Bautin ideal $I$. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs $ Arc(B_I\mathbb C^n,E)$. The example of planar quadratic vector fields in the Kapteyn normal form is further discussed., Comment: revised and significantly completed version, 3 figures are added

We consider arbitrary one-parameter cubic deformations of the Duffing oscillator $x"=x-x^3$. In the case when the first Melnikov function $M_1$ vanishes, but $M_2\neq 0$ we compute the general form of $M_2$ and study its zeros in a suitable complex domain., Comment: 17 pages, 5 figures, Bulletin des Sciences Math\'ematiques (1919) to appear. arXiv admin note: substantial text overlap with arXiv:1401.5419

The purpose of the present paper is to study the limit cycles of one-parameter perturbed plane Hamiltonian vector field $X_\varepsilon$ $$ X_\varepsilon : \left\{ \begin{array}{llr} \dot{x}=\;\; H_y+\varepsilon f(x,y)\\ \dot{y}=-H_x+\varepsilon g(x,y), \end{array} \;\;\;\;\; H~=\frac{1}{2} y^2~+U(x) \right. $$ which bifurcate from the period annuli of $X_0$ for sufficiently small $\varepsilon$. Here $U$ is a univariate polynomial of degree four without symmetry, and $f, g$ are arbitrary cubic polynomials in two variables. We take a period annulus and parameterize the related displacement map $d(h,\varepsilon)$ by the Hamiltonian value $h$ and by the small parameter $\varepsilon$. Let $M_k(h)$ be the $k$-th coefficient in its expansion with respect to $\varepsilon$. We establish the general form of $M_k$ and study its zeroes. We deduce that the period annuli of $X_0$ can produce for sufficiently small $\varepsilon$, at most 5, 7 or 8 zeroes in the interior eight-loop case, the saddle-loop case, and the exterior eight-loop case respectively. In the interior eight-loop case the bound is exact, while in the saddle-loop case we provide examples of Hamiltonian fields which produce 6 small-amplitude limit cycles. Polynomial perturbations of $X_0$ of higher degrees are also studied.

The cyclicity of the exterior period annulus of the asymmetrically perturbed Duffing oscillator is a well known problem extensively studied in the literature. In the present paper we provide a complete bifurcation diagram for the number of the zeros of the associated Melnikov function in a suitable complex domain., Comment: 11 pages, 5 figures

We prove that the cyclicity of a quadratic slow-fast integrable system of Darboux type with a double heteroclinic loop is finite and uniformly bounded., Comment: 13 figures, 15 pages

We prove that the number of limit cycles, which bifurcate from a two-saddle loop of a planar quadratic Hamiltonian system, under an arbitrary quadratic deformation, is less than or equal to three.

We prove that every heteroclinic saddle loop (a two-saddle cycle) occurring in an analytic finite-parameter family of plane analytic vector fields, may generate no more than a finite number of limit cycles within the family., Comment: 21 pages, 10 figures, a new section explaining the so called "Petrov trick" in the context of the paper is added. The paper will appear in "Functional Analysis and Its Applications" (2013)

We prove that the holonomy group at infinity of the Painleve VI equation is virtually commutative., Comment: 21 pages, 3 figures, Added references, Corrected typos

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let k^a be the algebraic closure of k. For a solution y, Ly=0, we determine the linear differential operator of minimal degree M and coefficients in k^a, such that My=0. This result is then applied to some Picard-Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka-Volterra type.

We find an upper bound to the maximal number of limit cycles, which bifurcate from a hamiltonian two-saddle loop of an analytic vector field, under an analytic deformation., Comment: revised version : we added an Appendix The paper is an extended version of my talk on the conference "Singularites des champs de vecteurs du plan, bifurcations et applications" held in CIRM, Luminy (France), 11-15 May, 2009

We study the stratum in the set of all quadratic differential systems $\dot{x}=P_2(x,y), \dot{y}=Q_2(x,y)$ with a center, known as the codimension-four case $Q_4$. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincar\'e-Pontryagin-Melnikov integral $I$. We show that the orbits of the unperturbed system are elliptic curves, and $I$ is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center.

Let $\Pi$ be an open, relatively compact period annulus of real analytic vector field $X_0$ on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from $\Pi$ under a given multi-parameter analytic deformation $X_\lambda$ of $X_0$ is finite, provided that $X_0$ is either Hamiltonian, or generic Darbouxian vector field., Comment: 22 pages, 1 figure

We study the leading term of the holonomy map of a perturbed plane polynomial Hamiltonian foliation. The non-vanishing of this term implies the non-persistence of the corresponding Hamiltonian identity cycle. We prove that this does happen for generic perturbations and cycles, as well for cycles which are commutators in Hamiltonian foliations of degree two. Our approach relies on the Chen's theory of iterated path integrals which we briefly resume., Comment: 17 pages

Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral., Comment: 13 pages, no figures

We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce the maximal number of limit cycles. We compute the associated Poincare-Pontryagin-Melnikov functions whose zeros control the number of limit cycles. To illustrate our approach, we determine the cyclicity of the annuli of two particular reversible systems., Comment: 28 pages

We prove that the maximal number of limit cycles which bifurcate from an open period annulus under a given multi-parameter analytic deformation of a given analytic vector field is the same as in an appropriate one-parameter analytic deformation of the field, provided that this cyclicity is finite. Along the same lines we give also a bound of the cyclicity of homoclinic saddle loops., Comment: 14 pages, 1 figure

We study the analogue of the infinitesimal 16th Hilbert problem in dimension zero. Lower and upper bounds for the number of the zeros of the corresponding Abelian integrals (which are algebraic functions) are found. We study the relation between the vanishing of an Abelian integral $I(t)$ defined over $\mathbb{Q}$ and its arithmetic properties. Finally, we give necessary and sufficient conditions for an Abelian integral to be identically zero., Comment: 21 pages

We classify all functions satisfying non-trivial families of PVI equations. It turns out that, up to an Okamoto equivalence, there are exactly four families parameterized by affine planes or lines. Each affine space is generated by points of "geometric origin", associated either to deformations of elliptic surfaces with four singular fibers, or to deformations of three-sheeted covers of the projective line with branching locus consisting of four points., Comment: 25 pages, 5 tables and 2 figures. The overall presentation is improved, the solution 1A is added to the main theorem, 3 figures and one table are removed (to save space)

We describe two algebraic solutions of the sixth Painlev\'e equation which are related to (isomonodromic) deformations of Picard-Fuchs equations of order two., Comment: 9 pages

We prove that the higher order Poincare-Pontryagin functions associated to the perturbed polynomial foliation $df -\epsilon (P dx + Q dy) =0$ satisfy a differential equation of Fuchs type., Comment: This is a revised version containing the following changes: reference [18] is added, some notations are changed, the proof of Theorem 1 is corrected, Theorem B1 and B2 are replaced now by Theorem B, some signs are corrected in the Appendix. The paper will appear in "Annales de la Faculte des Sciences de Toulouse", 2005

We study the displacement map associated to small one-parameter polynomial unfoldings of polynomial Hamiltonian vector fields on the plane. Its leading term, the generating function $M(t)$, has an analytic continuation in the complex plane and the real zeroes of $M(t)$ correspond to the limit cycles bifurcating from the periodic orbits of the Hamiltonian flow. We give a geometric description of the monodromy group of $M(t)$ and use it to formulate sufficient conditions for $M(t)$ to satisfy a differential equation of Fuchs or Picard-Fuchs type. As examples, we consider in more detail the Hamiltonian vector fields $\dot{z}=i\bar{z}-i(z+\bar{z})^3$ and $\dot{z}=iz+\bar{z}^2$, possessing a rotational symmetry of order two and three, respectively. In both cases $M(t)$ satisfies a Fuchs-type equation but in the first example $M(t)$ is always an Abelian integral (that is to say, the corresponding equation is of Picard-Fuchs type) while in the second one this is not necessarily true. We derive an explicit formula of $M(t)$ and estimate the number of its real zeroes.}, Comment: This is a revised verion containing 36 pages and 8 figures

Let $P(x)$ be a real polynomial of degree $2g+1$, $H=y^2+P(x)$ and $\delta(h)$ be an oval contained in the level set $\{H=h\}$. We study complete Abelian integrals of the form $$I(h)=\int_{\delta(h)} \frac{(\alpha_0+\alpha_1 x+... + \alpha_{g-1}x^{g-1})dx}{y}, h\in \Sigma,$$ where $\alpha_i$ are real and $\Sigma\subset \R$ is a maximal open interval on which a continuous family of ovals $\{\delta(h)\}$ exists. We show that the $g$-dimensional real vector space of these integrals is not Chebyshev in general: for any $g>1$, there are hyperelliptic Hamiltonians $H$ and continuous families of ovals $\delta(h)\subset\{H=h\}$, $h\in\Sigma$, such that the Abelian integral $I(h)$ can have at least $[\frac32g]-1$ zeros in $\Sigma$. Our main result is Theorem \ref{main} in which we show that when $g=2$, exceptional families of ovals $\{\delta(h)\}$ exist, such that the corresponding vector space is still Chebyshev.

We state two recent results concerning the linearization of integrable systems on generalised Jacobians. Then we apply this to the (complexified) spherical pendulum., Comment: to appear in CRM Proceedings and Lecture Notes (Vadim B. Kuznetsov, Ed.) Montreal, 2002

Let $(x(t),y(t))^\top$ be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form $P(t)x(t)+Q(t)y(t)$, where $P,Q$ are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behaviour of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeroes of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a nonoscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane., Comment: preprint of the University of Toulouse 3, November 2001

Consider an ordinary differential equation which has a Lax pair representation A'(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only onA(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold {A(x): det(A(x)-y I)= P(x,y)} of this Lax pair is an affine part of a non-compact commutative algebraic group---the generalized Jacobian of the spectral curve {(x,y): P(x,y)=0} with its points at "infinity" identified. Moreover, for suitable B(x), the Hamiltonian vector field defined by the Lax pairon the generalized Jacobian is translation--invariant. We provide two examples in which the above result applies., Comment: 20 pages, LaTex2e, to appear in Math. Zeitschrift

We prove that the heavy symmetric top (Lagrange, 1788) linearizes on a two-dimensional non-compact algebraic group -- the generalized Jacobian of an elliptic curve with two points identified. This leads to a transparent description of its complex and real invariant level sets. We also deduce, by making use of a Baker-Akhiezer function, simple explicit formulae for the general solution of Lagrange top., Comment: LaTex, 26 pages