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Iliya D. Iliev

Publications and source records attributed to Iliya D. Iliev.

13 recordsLinked to original sources
arXiv2022

The limit cycles in a generalized Rayleigh-Liénard oscillator

We compute the cyclicity of open period annuli of the following generalized Rayleigh-Liénard equation $$\ddot{x}+ax+bx^3-(λ_1+λ_2 x^2+λ_3\dot{x}^2+λ_4 x^4+λ_5\dot{x}^4+λ_6 x^6)\dot{x}=0$$ and the equivalent planar system $X_λ$, where the coefficients of the perturbation $λ_j$ are independent small parameters and $a, b$ are fixed nonzero constants. Our main tool is the machinery of the so called higher-order Poincaré-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 $\varepsilon \to λ(\varepsilon)$ and explicitly compute all possible Melnikov functions $M_n$ related to the deformation $X_{ λ(\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 $λ\to X_λ$, 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_{ λ} $ to the study of one-parameter deformations $X_{ λ(\varepsilon)} $.

math.DS↗
arXiv2019

On the cubic perturbations of the symmetric 8-loop Hamiltonian

We study arbitrary cubic perturbations of the symmetric 8-loop Hamiltonian, which are linear with respect to the small parameter. It is shown that when the first 4 coefficients in the expansion of the displacement functions corresponding to both period annuli inside the loop vanish, the system becomes integrable with the following three strata in the center manifold: Hamiltonian, reversible in $y$ and reversible in $x$. In the latter case, the first integral is of Darboux type and we calculate it explicitly. Next we prove that the cyclicity of each of period annuli inside the loop is five and the total cyclicity of both is at most nine. For this, we use Abelian integrals method together with careful study the geometry of the separatrix solutions of related Riccati equations in connection to the second-order Melnikov functions.

math.CA↗
arXiv2014

Cubic perturbations of elliptic Hamiltonian vector fields of degree three

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.

math.DS↗
arXiv2010

Non-uniform continuity of periodic Holm-Staley b-family of equations

We consider a family of non-evolutionary partial differential equations known as Holm - Staley b - family which includes the integrable Camassa-Holm and Degasperis-Procesi equations. We show that the solution map is not uniformly continuous. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

nlin.SI↗
arXiv2008

Quadratic perturbations of quadratic codimension-four centers

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.

math.DS↗
arXiv2007

Perturbations of quadratic centers of genus one

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.

math.DS↗
arXiv2004

The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields

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.}

math.DS↗
arXiv2002

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 $$I(h)=\int_{δ(h)} \frac{(α_0+α_1 x+... + α_{g-1}x^{g-1})dx}{y}, h\in Σ,$$ 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.

math.DS↗
arXiv2001

Two dimensional Fuchsian systems and the Chebyshev property

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.

math.DS↗