arXiv subjects
Publications and source records attributed to Lubomir Gavrilov.
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.
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 Poincaré-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 $Π$ 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 $Π$ under a given multi-parameter analytic deformation $X_λ$ of $X_0$ is finite, provided that $X_0$ is either Hamiltonian, or generic Darbouxian vector field.
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.
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.
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.
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.
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.
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.}
We prove that the higher order Poincare-Pontryagin functions associated to the perturbed polynomial foliation $df -ε(P dx + Q dy) =0$ satisfy a differential equation of Fuchs type.
We describe two algebraic solutions of the sixth Painlevé equation which are related to (isomonodromic) deformations of Picard-Fuchs equations of order two.
Let $P(x)$ be a real polynomial of degree $2g+1$, $H=y^2+P(x)$ and $δ(h)$ be an oval contained in the level set $\{H=h\}$. We study complete Abelian integrals of the form $$I(h)=\int_{δ(h)} \frac{(α_0+α_1 x+... + α_{g-1}x^{g-1})dx}{y}, h\in Σ,$$ where $α_i$ are real and $Σ\subset \R$ is a maximal open interval on which a continuous family of ovals $\{δ(h)\}$ 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 $δ(h)\subset\{H=h\}$, $h\inΣ$, such that the Abelian integral $I(h)$ can have at least $[\frac32g]-1$ zeros in $Σ$. Our main result is Theorem \ref{main} in which we show that when $g=2$, exceptional families of ovals $\{δ(h)\}$ 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.
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.
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.
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.