Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems

DOI10.1090/S1079-6762-99-00061-XzbMath0922.58076OpenAlexW1566141765MaRDI QIDQ4252962

Sergey Yu. Yakovenko, Dmitri Novikov

Publication date: 23 June 1999

Published in: Electronic Research Announcements of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://eudml.org/doc/223357

zbMATH Keywords

tangential Hilbert 16th problemdifferential algebraic generalizationisolated ovals of algebraic level curves

Mathematics Subject Classification ID

Topological structure of integral curves, singular points, limit cycles of ordinary differential equations (34C05) Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) (30C15) Applications of dynamical systems (37N99) Analytic theory of abelian varieties; abelian integrals and differentials (14K20) Bifurcations of limit cycles and periodic orbits in dynamical systems (37G15) Ordinary differential equations in the complex domain (34M99)

Related Items (15)

Centennial History of Hilbert's 16th Problem ⋮ Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop. II ⋮ On the number of limit cycles bifurcating from a non-global degenerated center ⋮ A rigorous study of possible configurations of limit cycles bifurcating from a hyper-elliptic Hamiltonian of degree five ⋮ On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle ⋮ The cyclicity of period annuli of a class of quintic Hamiltonian systems ⋮ Polynomial bounds for the oscillation of solutions of Fuchsian systems ⋮ On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity ⋮ On the number of zeros of Abelian integrals. A constructive solution of the infinitesimal Hilbert sixteenth problem ⋮ Redundant Picard-Fuchs system for abelian integrals ⋮ Abelian integrals and limit cycles ⋮ Complete hyperelliptic integrals of the first kind and their non-oscillation ⋮ The monotonicity of the ratio of two Abelian integrals ⋮ Weak infinitesimal Hilbert's 16th problem ⋮ Bounded decomposition in the Brieskorn lattice and Pfaffian Picard--Fuchs systems for Abelian integrals

Cites Work

• Trajectories of polynomial vector fields and ascending chains of polynomial ideals
• Meandering of trajectories of polynomial vector fields in the affine \(n\)-space
• Bifurcation of planar vector fields and Hilbert's sixteenth problem
• The complement of the bifurcation variety of a simple singularity
• Non-oscillation of elliptic integrals
• On the number of zeros of analytic functions in a neighborhood of a Fuchsian singular point with real spectrum
• Simple exponential estimate for the number of real zeros of complete abelian integrals
• Double exponential estimate for the number of zeros of complete Abelian integrals and rational envelopes of linear ordinary differential equations with an irreducible monodromy group
• Some unsolved problems in the theory of differential equations and mathematical physics
• A class of Hamiltonian systems with increasing periods.
• On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields
• An explicit bound for the multiplicity of zeros of generic Abelian integrals
• Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians
• Complete Abelian integrals as rational envelopes
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