We prove that the lowest upper bound for the number of isolated zeros of the Abelian integrals associated to quadratic Hamiltonian vector fields having a center and an invariant straight line after quadratic perturbations is one.
Let be the real vector space of Abelian integralswhere is a fixed real polynomial, is an arbitrary real polynomial and , , is the interior of the oval of which surrounds the origin and tends to it as . We prove that if is a semiweighted homogeneous polynomial with only Morse critical points, then is a free finitely generated module over the ring of real polynomials , and compute its rank. We find the generators of in the case when is an arbitrary cubic polynomial. Finally we...
In this paper we consider the system of Hamiltonian differential equations, which determines small oscillations of a dynamical system with n parameters. We demonstrate that this system determines an affinor structure J on the phase space TRⁿ. If J² = ωI, where ω = ±1,0, the phase space can be considered as the biplanar space of elliptic, hyperbolic or parabolic type. In the Euclidean case (Rⁿ = Eⁿ) we obtain the Hopf bundle and its analogs. The bases of these bundles are, respectively, the projective...
We simplify and generalize McDuff’s construction of symplectic 4-manifolds with disconnected boundary of contact type in terms of the linking pairing defined on the dual of 3-dimensional Lie algebras. This leads us to an observation that an Anosov flow gives rise to a bi-contact structure, i.e. a transverse pair of contact structures with different orientations, and the construction turns out to work for 3-manifolds which admit Anosov flows with smooth invariant volume. Finally, new examples of...
We apply Gromov’s method of convex integration to problems related to the existence and uniqueness of symplectic and contact structures