scholarly journals Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

2000 ◽  
Vol 20 (6) ◽  
pp. 1671-1686 ◽  
Author(s):  
LUBOMIR GAVRILOV ◽  
ILIYA D. ILIEV
Keyword(s):  
Vector Field ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Field ◽  
Finite Plane ◽  
Order Analysis ◽  
Hamiltonian Vector Fields

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 Poincaré–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 Poincaré–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 ($n=2$) 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.

scholarly journals A Lie connection between Hamiltonian and Lagrangian optics

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Alex J. Dragt
Keyword(s):  
Vector Field ◽  
Lie Algebra ◽  
Vector Fields ◽  
Geometrical Optics ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Field ◽  
Third Order ◽  
Hamiltonian Vector Fields ◽  
Uniform Medium ◽  
International Audience

International audience It is shown that there is a non-Hamiltonian vector field that provides a Lie algebraic connection between Hamiltonian and Lagrangian optics. With the aid of this connection, geometrical optics can be formulated in such a way that all aberrations are attributed to ray transformations occurring only at lens surfaces. That is, in this formulation there are no aberrations arising from simple transit in a uniform medium. The price to be paid for this formulation is that the Lie algebra of Hamiltonian vector fields must be enlarged to include certain non-Hamiltonian vector fields. It is shown that three such vector fields are required at the level of third-order aberrations, and sufficient machinery is developed to generalize these results to higher order.

Weak Center and Bifurcation of Critical Periods in a Cubic Z2-Equivariant Hamiltonian Vector Field

2015 ◽  
Vol 25 (11) ◽  
pp. 1550143 ◽  
Author(s):  
Yusen Wu ◽  
Wentao Huang ◽  
Yongqiang Suo
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Local Bifurcation ◽  
Hamiltonian Vector Field ◽  
Critical Periods ◽  
Weak Center ◽  
Hamiltonian Vector Fields ◽  
Center Conditions ◽  
Bifurcation Of Critical Periods

This paper focuses on the problems of weak center and local bifurcation of critical periods for a class of cubic Z2-equivariant planar Hamiltonian vector fields. By computing the period constants carefully, one can see that there are three weak centers: (±1, 0) and the origin. The corresponding weak center conditions are also derived. Meanwhile, we address the problem of the coexistence of bifurcation of critical periods that occurred from (±1, 0) and the origin.

scholarly journals The number of limit cycles of polynomial deformations of a Hamiltonian vector field

1990 ◽  
Vol 10 (3) ◽  
pp. 523-529 ◽  
Author(s):  
P. Mardešić
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Upper Bound ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Field ◽  
Image Position

AbstractWe prove that the lowest upper bound for the number of limit cycles of small nonconservative polynomial deformations of degree n of the Hamiltonian vector field is n − 1.A consequence is that the lowest upper bound for the number of limit cycles of generic n-parameter deformations of cusps is n − 1.

scholarly journals A construction of a large family of commuting pairs of integrable symplectic birational four-dimensional maps

2017 ◽  
Vol 473 (2198) ◽  
pp. 20160535 ◽  
Author(s):  
Matteo Petrera ◽  
Yuri B. Suris
Keyword(s):  
Hamiltonian Systems ◽  
Symplectic Structure ◽  
Vector Fields ◽  
Large Family ◽  
Hamiltonian Vector ◽  
Discretization Scheme ◽  
Integrals Of Motion ◽  
Completely Integrable ◽  
Hamiltonian Vector Fields ◽  
Commuting Pairs

We give a construction of completely integrable four-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan–Hirota–Kimura discretization scheme, we arrive at pairs of birational four-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on R 4 , and possess two independent integrals of motion, which are perturbations of the original Hamilton functions and which are in involution with respect to the perturbed symplectic structure. Thus, these maps are completely integrable in the Liouville–Arnold sense. Moreover, under a suitable normalization of the original pairs of vector fields, the pairs of maps commute and share the invariant symplectic structure and the two integrals of motion.

scholarly journals Perturbations of a Hamiltonian family of cubic vector fields

1991 ◽  
Vol 44 (1) ◽  
pp. 139-147
Author(s):  
A.M. Urbina ◽  
M. Cañas ◽  
G. León de la Barra ◽  
M. León de la Barra
Keyword(s):  
Vector Field ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Vector Fields ◽  
Global Analysis ◽  
Hamiltonian Vector Field ◽  
Hamiltonian Graph ◽  
Polynomial Vector Fields ◽  
The Stability ◽  
Type Of Stability

This paper is related with the configurations of limit cycles for cubic polynomial vector fields in two variables (χ3).It is an open question to decide whether every limit cycle configuration in χ3 can be obtained by perturbation of a corresponding Hamiltonian configuration of centres and graphs.In this work, by considering perturbations of the Hamiltonian vector field XH = (Hy, − Hx), where H(x, y) = [a(x + h)2 + by2 − 1] [a(x − h)2 + by2 − 1], we make a global analysis of the possible cases.The vector field XH has three centres (C−, C+ and the origin) and two saddles. By means of quadratic perturbations the centres become fine foci where C−and C+ have the same type of stability but opposed to that one of the origin and infinity. Further introducing cubic perturbations changes the stability of C−, C+ and the cycle at infinity and generates limit cycles. Lastly extra linear terms change the stability of the origin and generate another limit cycle.Finally, we analyse the rupture of saddle connection of the Hamiltonian field under perturbation, via Melnikov's integral, in order to complete the study of the global phase portrait and to consider the possibility of new limit cycles emerging from the Hamiltonian graph.

scholarly journals ∞-jets of diffeomorphisms preserving orbits of vector fields

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Sergiy Maksymenko
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Homogeneous Polynomials ◽  
Local Flow ◽  
Linear Vector ◽  
Identity Component ◽  
Hamiltonian Vector Fields

AbstractLet F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = .In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O.We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.

scholarly journals THE g-AREAS AND THE COMMUTATOR LENGTH

2013 ◽  
Vol 24 (07) ◽  
pp. 1350057 ◽  
Author(s):  
FRANÇOIS LALONDE ◽  
ANDREI TELEMAN
Keyword(s):  
Vector Fields ◽  
Morse Function ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Field ◽  
Linear Combinations ◽  
Hamiltonian Diffeomorphisms ◽  
Hamiltonian Vector Fields ◽  
Commutator Length ◽  
Lie Brackets ◽  
Hamiltonian Diffeomorphism

The commutator length of a Hamiltonian diffeomorphism f ∈ Ham (M,ω) of a closed symplectic manifold (M,ω) is by definition the minimal k such that f can be written as a product of k commutators in Ham (M,ω). We introduce a new invariant for Hamiltonian diffeomorphisms, called the k+-area, which measures the "distance", in a certain sense, to the subspace [Formula: see text] of all products of k commutators. Therefore, this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of k commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of k Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed k, the set of linear combinations of k such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function f on a symplectic Riemann surface M (verifying a weak genericity condition) we describe the linear space of commutators of the form {f, g}, with [Formula: see text].

CENTER PROBLEM AND MULTIPLE HOPF BIFURCATION FOR THE Z6-EQUIVARIANT PLANAR POLYNOMIAL VECTOR, FIELDS OF DEGREE 5

2009 ◽  
Vol 19 (05) ◽  
pp. 1741-1749 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Vector Fields ◽  
Hamiltonian Vector Field ◽  
Center Problem ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Perturbed System ◽  
Lyapunov Constants

This paper proves that a Z6-equivariant planar polynomial vector field of degree 5 has at least six symmetric centers, if and only if it is a Hamiltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z6-equivariant perturbations, the conclusion that the perturbed system has at least 24 limit cycles with the scheme 〈 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4〉 is rigorously proved. Two schemes of distributions of limit cycles are given.

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