Real Polynomials Of Degree Research Articles

Maximum number of limit cycles of a second order differential system

In this paper, we study the limit cycles of a perturbed differential in $\mathbb{R} ^2$, given by \begin{equation*} \left\{ \begin{array}{ccl} \overset{.}{x} y ,\\ \overset{.}{y} -x-\epsilon (1+\sin^n (\theta) \cos^m (\theta))H(x,y), \end{array} \right. \end{equation*} where $\epsilon$ is a small parameter, $m$ and $n$ are non-negative integers, $\tan(\theta)=y/x$, and $P(x,y)$ is a real polynomial of degree $n \geq 1$. Using Averaging theory of first order we provide an upper bound for the maximum number of limit cycles. Also, we provide some examples to confirm and illustrate our results.

ON THE NUMBER OF NONTRIVIAL RATIONAL SOLUTIONS FOR ABEL EQUATIONS

In this paper, a systematic algorithm is provided to determine the sharp upper bound on the number of nontrivial rational solutions for the Abel differential equations $dy/dx=f_m(x)y^2+g_n(x)y^3$, where $f_m(x)$ and $g_n(x)$ are real polynomials of degree $m$ and $n$ respectively. As an application, we present a thorough study for an important case, $(m,n)=(4,9$).

On the maximum number of limit cycles of a planar differential system

In this work, we are interested in the study of the limit cycles of a perturbed differential system in (mathbb{R}^2), given as follows[left{begin{array}{l}dot{x}=y, \dot{y}=-x-varepsilon (1+sin ^{m}(theta ))psi (x,y),%end{array}%right.]where (varepsilon) is small enough, (m) is a non-negative integer, (tan (theta )=y/x), and (psi (x,y)) is a real polynomial of degree (ngeq1). We use the averaging theory of first-order to provide an upper bound for the maximum number of limit cycles. In the end, we present some numerical examples to illustrate the theoretical results.

Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

In this paper we deal with the Lienard system $$\dot{x}=y, \dot{y}=-f_m(x)y-g_n(x),$$ where $$f_m(x)$$ and $$g_n(x)$$ are real polynomials of degree m and n, respectively. We call this system the Lienard system of type (m, n). For this system, we proved that if $$m+1\le n\le [\frac{4m+2}{3}]$$ , then the maximum number of hyperelliptic limit cycles is $$n-m-1$$ , and this bound is sharp. This result indicates that the Lienard system of type $$(m,m+1)$$ has no hyperelliptic limit cycles. Secondly, we present examples of irreducible algebraic curves of arbitrary high degree for Lienard systems of type $$(m,2m+1)$$ . Moreover, these systems have a rational first integral. Finally, we proved that the Lienard system of type (2, 5) has at most one hyperelliptic limit cycle, and this bound is sharp.

THE NUMBER OF RATIONAL SOLUTIONS OF ABEL EQUATIONS

<abstract> In this paper, we study rational solutions of the Abel differential equations $ dy/dx = f_m(x)y^2+g_n(x)y^3 $, where $ f_m(x) $ and $ g_n(x) $ are real polynomials of degree $ m $ and $ n $ respectively. The main result of the paper is as follows: We give a systematic upper bound on the number of the nontrivial rational solutions of such equations in all these cases. Then we prove that these upper bounds can be reached in most cases. Finally, we present some examples of Abel equations having exactly $ i $ nontrivial rational solutions, where $ 1\leq i\leq 5 $. </abstract>

On the Number of Hyperelliptic Limit Cycles of Liénard Systems

In this paper, we study the maximum number, denoted by $H(m,n)$, of hyperelliptic limit cycles of the Li\'enard systems $$\dot x=y, \qquad \dot y=-f_m(x)y-g_n(x),$$ where, respectively, $f_m(x)$ and $g_n(x)$ are real polynomials of degree $m$ and $n$, $g_n(0)=0$. The main results of the paper are as follows: In term of $m$ and $n$ of the system, we obtain the upper bound and lower bound of $H(m,n)$ in all the possible cases. Furthermore, these upper bound can be reached in some cases.

On Hilbert’s 17th problem in low degree

Artin solved Hilbert's 17th problem, proving that a real polynomial in $n$ variables that is positive semidefinite is a sum of squares of rational functions, and Pfister showed that only $2^n$ squares are needed. In this paper, we investigate situations where Pfister's theorem may be improved. We show that a real polynomial of degree $d$ in $n$ variables that is positive semidefinite is a sum of $2^n-1$ squares of rational functions if $d\leq 2n-2$. If $n$ is even, or equal to $3$ or $5$, this result also holds for $d=2n$.

Limit cycles of the classical Liénard differential systems: A survey on the Lins Neto, de Melo and Pugh’s conjecture

In 1977 Lins Neto et al. (1977) conjectured that the classical Liénard system ẋ=y−F(x),ẏ=−x, with F(x) a real polynomial of degree n, has at most [(n−1)/2] limit cycles, where [⋅] denotes the integer part function. In this paper we summarize what is known and what is still open on this conjecture. For the known results on this conjecture we present a complete proof.

Tangential Center Problem for a Family of Non-generic Hamiltonians

The tangential center problem was solved by Yu. S. Ilyashenko in the generic case Mat Sbornik (New Series), 78, 120, 3,360–373, (1969). With the aim of having well-understood models of non-generic Hamiltonians, we consider here a family of non-generic Hamiltonians, whose Hamiltonian is of the form \(F=\prod f_{j}\), where fj are real polynomials of degree ≥ 1. For this family, the genericity assumption of transversality at infinity fails and the coincidence of the critical values for different critical points is allowed. We consider some geometric conditions on these polynomials in order to compute the orbit under monodromy of their vanishing cycles. Under those conditions, we provide a solution of the tangential center problem for this family.

On the Exact Number of Zeros of Certain Even Degree Polynomails : A Sharkovsky Theorem Approach

The problem of finding or characterizing the zeros of polynomial functions has a long history. The obtained results varied from theory, algorithms and numerical simulations. Historically, this study began with the fundamental theorem of algebra proved by Gauss. The most known results in this direction is the fact that a real polynomial of degree n has at most n real zeros. There is also the so called Descartes rule of signs concerning the number of positive zeros. Some generalizations of Descartes rule are know such as the BudanFourier theorem that gives an upper bound for the number of zeros of a polynomial. Also, the Sturm’s theorem that gives a method for determining the exact number of zeros in an interval [Hen, chapter 6] and [Hou, chapter 2]. Recent results uses the classical Enestrom-Kakeya theorem to restricts the location of the zeros based on a condition imposed on the coefficients of the polynomial under invistigation. See [Bre] and references therain. In this paper, we will use a dynamical system result concerning periodic points of a continuous function. The result is called Sharkovsky theorem [Sha1, Sha2, Sha3, Sha4, Sha5] that gives a complete description of possible sets of periods for continuous mappings defined on an interval. The interval need not be closed or bounded. The main idea used here is the notion of

Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center

In this paper, we deal with the following differential system [Formula: see text] where p, q are positive integers, and P(x, y), Q(x, y) are real polynomials of degree n, we obtain an upper bound for the maximum number of limit cycles bifurcating from the period annulus of a quasi-homogeneous center, that is (n - 1)p1 + (t + 1)q - 1 + 2rp1q1(q + 3) + 2tqrp1q1, where t = [n/2q] + 2, (p, q) = r(p1, q1), p1 and q1 are coprime.

A generalization of Descartes’ rule of signs and fundamental theorem of algebra

Abstract Descartes’ rule of signs yields an upper bound for the number of positive and negative real roots of a given polynomial. The fundamental theorem of algebra implies a similar property; every real polynomial of degree n ⩾ 1 has at most n real zeroes. In this paper, we describe axiomatically function families possessing one or another of these properties. The resulting families include, at least, all polynomial functions and sums of exponential functions. As an application of our approach, we consider, among other things, a method for identifying certain type of bases for the Euclidean space.

Maximal univalent disks of real rational functions and Hermite-Biehler polynomials

The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k has all roots in the open upper half-plane if and only if s=p+iq where p and q are real polynomials of degree k and k-1 resp. with all real, simple and interlacing roots, and q has a negative leading coefficient. Considering roots of p as cyclically ordered on RP^1 we show that the open disk in CP^1 having a pair of consecutive roots of p as its diameter is the maximal univalent disk for the function R=\frac{q}{p}. This solves a special case of the so-called Hermite-Biehler problem.

Topological classification of morse polynomials

The topological classification is discussed for real polynomials of degree 4 in two real independent variables whose critical points and critical values are all different. It is proved that among the 17 746 topological types of smooth functions with the same number of critical points, at most 426 types are realizable by polynomials of degree 4.

THE MIKHAILOV STABILITY CRITERION REVISITED

It is shown that the principle of the argument is the basis for the Mikhailov’s stability criterion for linear continuous systems. Mikhailov’s criterion states that a real Hurwitz polynomial ) (s  of degree n satisfies the monotonic phase increase, that is to say the argument of ) ( jw  goes through n quadrants as w runs from zero to infinity. In this paper, the generalized Mikhailov criterion where a real polynomial of degree n with no restriction on the roots location is considered. A method based on the argument is used to determine the number of roots in each half of the splane as well as on the imaginary axis if any.

A realization theorem in the context of the Schur-Szegő composition

Every real polynomial of degree n in one variable with root −1 can be represented as the Schur-Szegő composition of n − 1 polynomials of the form (x + 1) n−1(x + a i ), where the numbers a i are uniquely determined up to permutation. Some a i are real, and the others form complex conjugate pairs. In this note, we show that for each pair (ρ, r), where 0 ⩽ ρ, r ⩽ [n/2], there exists a polynomial with exactly ρ pairs of complex conjugate roots and exactly r complex conjugate pairs in the corresponding set of numbers a i .

Complex asymptotics of Poincaré functions and properties of Julia sets

AbstractThe asymptotic behaviour of the solutions of Poincaré's functional equation f(λz) = p(f(z)) (λ &gt; 1) for p a real polynomial of degree ≥ 2 is studied in angular regions W of the complex plain. It is known [9, 10] that f(z) ~ exp(zρF(logλz)), if f(z) → ∞ for z ∞ and z ∈ W, where F denotes a periodic function of period 1 and ρ = logλ deg(p). In this paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function F is characterised in terms of geometric properties of the Julia set of p. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of f is related to the harmonic measure on the Julia set of p.

Block orthogonal polynomials: II. Hermite and Laguerre standard block orthogonal polynomials

The standard block orthogonal (SBO) polynomials Pi;n(x), 0 ⩽ i ⩽ n are real polynomials of degree n which are orthogonal with respect to a first Euclidean scalar product to polynomials of degree less than i. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. Applying the general results obtained in a previous paper, we determine and investigate these polynomials when the first scalar product corresponds to Hermite (resp. Laguerre) polynomials. These new sets of polynomials, which we call Hermite (resp. Laguerre) SBO polynomials, provide a basis of functional spaces well suited for some applications requiring to take into account special linear constraints which can be recast into an Euclidean orthogonality relation.

Anisotropic real curves and bordered line arrangements

Let X be a smooth projective real algebraic curve of genus g greater than 1. The canonical divisor on X defines a morphism k : X→Pg−1 from X into real projective space Pg−1. The image k(X) of k is isomorphic either to X , or to the real projective line P1, or to the “empty circle” S1, understood as the smooth projective rational curve given by the affine equation x2 + y2 = −1. The real curve X is said to be anisotropic if k(X) is isomorphic to the empty circle. Let X be an anisotropic curve. The following facts are known [Gross and Harris 1981, Proposition 6.2]: the genus g of X is odd, its real locus X (R) is empty, and X may be represented, in affine 3-space, by a pair of equations of the form x2+ y2 = −1 and w2= p(x, y), where p is a real polynomial of degree g+1. The canonical morphism k can then be identified with the projection X 3 (w, x, y) 7→ (x, y). The main problem in studying anisotropic curves is the absence of real points both in X and in k(X). To circumvent this, we proceed as follows. Let X be defined by equations as above. We can consider the double covering of the plane, ramified along the g+ 1 real lines joining the g+ 1 pairs of complex conjugated branch points of k in S1. This is a real surface D having real points, and turns out to be tantamount to and easier to study than the original curve X . Our results concern the moduli spaces of anisotropic curves. More precisely, we prove that there is an isomorphism between the moduli space of anisotropic curves and the moduli space of double coverings of P2 ramified along real line arrangements (see Theorem 7.1). As a corollary, we improve Proposition 6.2 of [Gross and Harris 1981], by showing, among other things, that any anisotropic

CERTAIN REPRESENTATION OF A REAL POLYNOMIAL

We show that every real polynomial of degree n can be represented by the sum of two real polynomials of degree n, each having only real zeros. As an example of this, we consider a real polynomial with even degree whose all zeros lie on the imaginary axes except the origin.