We study the Hess-Appelrot case of the Euler-Poisson system which describes dynamics of a rigid body about a fixed point. The first one is represented by M. In covering these and other topics, this book underlines the unifying role of the monogropy group. In the theory of homomorphic foliations there appear the Ecalle-Voronin-Martinet-Ramis moduli. The E-mail message field is required. The obtained results are used in a direct derivation of the Deprits formula for the isoenergetic degeneracy determinant in the restricted three—body problem. In particular, we prove existence and uniqueness of an invariant torus which appears in this family.
The material is addressed to a wide audience, ranging from specialists in the theory of ordinary differential equations to algebraic geometers. We show that before perturbation there exists a one-parameter family of periodic solutions obtained via a nondegenarate Hopf bifurcation and after perturbation there remains at most one limit cycle of small amplitude and bounded period. Register a Free 1 month Trial Account. We improve the known results about finite cyclicity of the graphics H 81 , H 101 , I 42 by showing that their cyclicities are equal to 2. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures.
The material is addressed to a wide audience, ranging from specialists in the theory of ordinary differential equations to algebraic geometers. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We consider Hamiltonian autonomous systems with n degrees of freedom near a singular point. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. It turns out that the normal hyperbolicity takes place only in the case of 1 : q resonance. The new system has an invariant surface close to L and the system restricted to it has a certain number of limit cycles.
The linearized in ε condition for limit cycles leads to the notion of generalized Abelian integrals. We show that the cyclicity of each of the first three graphics is equal to 2 and that the cyclicity of the fourth one is equal to three. Readers can quickly get introduced to modern and vital mathematical theories, such as singularity theory, analytic theory of ordinary differential equations, holomorphic foliations, Galois theory, and parts of algebraic geometry, without searching in vast literature. We present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case. We prove that the extended Hamiltonian systems do not have any additional algebraic first integral besides the known special cases of the third and fifth Painlevé equations. In the proofs we use equations in variations with respect to a parameter and Liouville's theory of elementary functions.
In the theory of systems of linear differential equations one has the Riemann-Hilbert problem, the Stokes phenomena and the hypergeometric functions with their multidimensional generalizations. We show that before perturbation there exists a one-parameter family of periodic solutions obtained via a nondegenarate Hopf bifurcation and after perturbation there remains at most one limit cycle of small amplitude and bounded period. The first one is represented by Singer's theorem about integration of polynomial planar vector fields and the second is Khovanskii's monodromy group of a multivalued analytic first integral of such vector field. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. In the case of generalized cusp we present the complete formal orbital normal form; it contains functional moduli.
We find four components modulo the action of A3 of the center variety, i. The technique used is a refinement of the technique of Khovanskii as adapted to the finite cyclicity of graphics by Il'yashenko and Yakovenko, together with an equivalent of the Bautin trick to treat the center case. We present some unexpected relations between geometrical properties of the curves ϕ,ψ and singularities of the map Expan. It is known that this number is finite for an individual vector field. We present a connection between the differential Galois theory and topological Galois theory.
In the case of absence of resonances of order less than or equal to 4 we present a direct computation of the Birkhoff normal form. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. Finally, we present formal classification of nilpotent singularities and prove analyticity of the Takens prenormal form. Now take a polynomial perturbation of this system. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. This implies that f3 x satisfies a sixth order linear differential equation with irregular singularity at infinity. We study orbital normal forms for analytic planar vector fields with nilpotent singularity.
Simon Stevin 15 927—34 a generalization of the Takens normal form for a nilpotent singularity of a vector field in was obtained. The book contains a lot of results which are usually spread in many sources. We also study the situation near the intersection of different components of the center variety. We estimate this number of zeros by C 1+C 2 n, where n is the degree of the perturbation. Holomorphic foliations : global aspects -- 11. We show that the monodromy maps together with certain additional maps, which are analogues of the Stokes operators, generate a solvable group.