Iliya D. Iliev

Perturbations of quadratic centers

Abstract We study the bifurcation of limit cycles in general quadratic perturbations of plane quadratic vector fields having a center at the origin. For any of the cases, we determine the essential perturbation and compute the corresponding bifurcation function. As an application, we find the precise location of the subset of centers in Q 3 R surrounded by period annuli of cyclicity at least three. Two specific cases are considered in more detail: the isochronous center S 1 and one of the intersection points ( Q 4 + ) of Q 4 and Q 3 R . We prove that the period annuli around S 1 and Q 4 + have cyclicity two and three respectively. The proof is based on the possibility to derive appropriate Picard-Fuchs equations satisfied by the independent integrals included in the related bifurcation function.

Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians

An explicit upper bound is derived for the number of the zeros of the integral of degree n polynomials f, g, on the open interval for which the cubic curve contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals , i,j=0,1, generating the module of complete Abelian integrals I(h) (over the ring of polynomials in h).

Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops

We investigate the bifurcations of limit cycles in a class of planar quadratic integrable (non-Hamiltonian) systems with two centres, both surrounded by unbounded heteroclinic loops, under small quadratic perturbations. By a careful study of the number of zeros of Abelian integrals based on the geometric properties of some planar curves, defined by ratios of such integrals, we obtain complete results about the number and the distribution of limit cycles bifurcating from the two period annuli.

The number of limit cycles due to polynomial perturbations of the harmonic oscillator

We consider arbitrary polynomial perturbations formula here of the harmonic oscillator. In (1), f and g are polynomials of x , y with coefficients depending analytically on the small parameter e. Let us denote n = max (deg f , deg g ), H = ½( x 2 + y 2 ). Using the energy level H = h as a parameter, we can express the first return mapping of (1) in terms of h and e. For the corresponding displacement function d ( h , e) = [Pscr ]( h , e)− h we obtain the following representation as a power series in e: formula here which is convergent for small e. The Melnikov functions M k ( h ) are defined for h [ges ]0. Each isolated zero h 0 ∈ (0, ∞) of the first non-vanishing coefficient in (2) corresponds to a limit cycle of (1) emerging from the circle x 2 + y 2 = 2 h 0 when e increases from zero. Our main result in this paper is the following.

Complete hyperelliptic integrals of the first kind and their non-oscillation

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) = ∫ δ(h) (α 0 + α 1 x +... + α g-1 x g-1 /y)dx y, h ∈ Σ, where α i are real and E ⊂ 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) ⊂ {H = h}, h E Σ, such that the Abelian integral I(h) can have at least [3/2g] - 1 zeros in Σ. Our main result is Theorem 1 in which we show that when g = 2, exceptional families of ovals {δ(h)} exist, such that the corresponding vector space is still Chebyshev.

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

We study degree n polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincare-Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is n 1. In the present paper we prove that if the first Poincare-Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is 2 .n 1/. In the case when the perturbation is quadratic (n D 2) we obtain a complete result—there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.

Quadratic perturbations of quadratic codimension-four centers

Abstract We study the stratum in the set of all quadratic differential systems x ˙ = P 2 ( x , y ) , 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 Poincare–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 Petrovs 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.

Stability of periodic travelling shallow-water waves determined by Newton's equation

We study the existence and stability of periodic travelling-wave solutions for generalized Benjamin–Bona–Mahony and Camassa–Holm equations. To prove orbital stability, we use the abstract results of Grillakis–Shatah–Strauss and the Floquet theory for periodic eigenvalue problems.

Two-dimensional Fuchsian systems and the Chebyshev property ☆

Let (x(t),y(t))⊤ 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 behavior 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 zeros 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 non-oscillation 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.

On the limit cycles available from polynomial perturbations of the Bogdanov-Takens Hamiltonian

The displacement map related to small polynomial perturbations of the planar Hamiltonian systemdH=0 is studied in the elliptic caseH=1/2y2+1/2x2−1/3x3. An estimate of the number of isolated zeros for each of the successive Melnikov functionsMk(h),k=1, 2,…is given in terms of the orderk and the maximal degreen of the perturbation. This sets up an upper bound to the number of limit cycles emerging from the periodic orbits of the Hamiltonian system under polynomial perturbations.