The Hamiltonian system corresponding to the (generalized) Hénon–Heiles H= 1/2(px2+py2)+1/2Ax2+1/2By2+x2y+εy3 is known be integrable in following three cases: (A=B, ε=1/3); (ε=2); (B=16A, ε=16/3). In first two has been integrated by making use of genus one and theta functions. We show that third case can also elliptic Finally, using Fairbanks theorem, we find Lax pairs for each systems under consideration.

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

It is shown that, for a planar Hamiltonian quadratic system with center, the period of associated periodic orbits strictly increasing function energy.

Nous montrons que toute fonction de Poincare-Pontryagin d'ordre superieur, associee au feuilletage polynomial perturbe defini par df - e (Pdx + Qdy) = 0, verifie une equation fuchsienne.

We study the displacement map associated to small one-parameter polynomial unfoldings of Hamiltonian vector fields on plane. Its leading term, generating function M ( t ), has an analytic continuation in complex plane and real zeroes ) correspond limit cycles bifurcating from periodic orbits flow. give a geometric description monodromy group use it formulate sufficient conditions for satisfy differential equation Fuchs or Picard-Fuchs type. As examples, we consider more detail...

We study isochronous centres of plane polynomial Hamiltonian systems, and more generally, Morse critical points complex functions. Our first result is that if the function H a non-degenerate semi-weighted homogeneous polynomial, then it cannot have an point, unless associate system linear, to say degree two. second gives topological obstruction for isochronicity. Namely, let be continuous family one-cycles contained in level set , vanishing at point H, as . prove good with only simple...

where H(x,y)=(x 2 +y )/2+... is a fixed real polynomial, R(x,y) an arbitrary polynomial and {H≤h}, h∈[0,h ˜], the interior of oval H which surrounds origin tends to it as h→0. We prove that if H(x,y) semiweighted homogeneous with only Morse critical points, then 𝒜 free finitely generated module over ring polynomials ℝ[h], compute its rank. find generators in case when cubic polynomial. Finally we apply this study degree n perturbations quadratic reversible Hamiltonian vector fields one...

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a real polynomial of degree alttext="2 g plus 1"> <mml:mn>2</mml:mn>...

Abstract Let Π be an open period annulus of a plane analytic vector field X 0 . We prove that the maximal number limit cycles which bifurcate from under given multi-parameter deformation λ is same as in appropriate one-parameter (ε) , provided this cyclicity finite. Along lines, we also give bound for homoclinic saddle loops.

We compute the cyclicity of open period annuli following generalized Rayleigh-Liénard equation \begin{document}$ \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 $\end{document} and equivalent planar system $ X_\lambda $, where coefficients perturbation \lambda_j are independent small parameters a, b fixed nonzero constants. Our main tool is machinery so called higher-order Poincaré-Pontryagin-Melnikov functions (Melnikov...

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

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

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

We prove that the cyclicity of a slow-fast integrable system Darboux type with double heteroclinic loop is finite and uniformly bounded.