Upper Bounds of Topology of Complex Polynomials in Two Variables

Glutsyuk, Alexey

doi:10.17323/1609-4514-2005-5-4-781-828

Cited by 5 publications

(10 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similar to the dissipative case, we use the normal bundle NT loc of T loc , where NT μ,ν loc ∼ = T n × R n × R 2q for each (μ, ν), to simplify the problem, see Section 3.1. However, the scaling operator D ε which defines normal linearization, slightly differs: D ε (x, y loc , z) = x, ε 2 y loc , εz , for ε > 0, (41) see [26,Section 2a,p. 9].…”

Section: Normal Linear Stability: Integrable Symplectic Casementioning

confidence: 99%

See 1 more Smart Citation

Normal linear stability of quasi-periodic tori

Broer

Hoo

Naudot

2007

Journal of Differential Equations

113

View full text Add to dashboard Cite

We consider families of dynamical systems having invariant tori that carry quasi-periodic motions. Our interest is the persistence of such tori under small, nearly-integrable perturbations. This persistence problem is studied in the dissipative, the Hamiltonian and the reversible setting, as part of a more general KAM theory for classes of structure preserving dynamical systems. This concerns the parametrized KAM theory as initiated by Moser [J.K. Moser, On the theory of quasiperiodic motions, SIAM Rev. 8 (2) (1966)145-172; J.K. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967) 136-176] and further developed in [G.B. Huitema, Unfoldings of quasi-periodic tori, PhD thesis, University of Groningen, 1988; H.W. Broer, G.B. Huitema, F. Takens, Unfoldings of quasi-periodic tori, Mem. Amer. Math. Soc. 83 (421) (1990) 1-82; H.W. Broer, G.B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, J. Dynam. Differential Equations 7 (1) (1995) 191-212]. The corresponding nondegeneracy condition involves certain (trans-)versality conditions on the normal linear, leading, part at the invariant tori. We show that as a consequence, a Cantor family of Diophantine tori with positive Hausdorff measure is persistent under nearly-integrable perturbations. This result extends the above references since presently the case of multiple Floquet exponents is included. Our leading example is the normal 1 : −1 resonance, which occurs a lot in applications, both Hamiltonian and reversible. As an illustration of this we briefly describe the Lagrange top coupled to an oscillator.

show abstract

Section: Normal Linear Stability: Integrable Symplectic Casementioning

confidence: 99%

“…A proof of Theorem A.1 is given now. It is closely related to [41] and based on application of the Rouché lemma to the characteristic polynomial of Ω(μ).…”

Section: Remarks 52mentioning

confidence: 99%

Normal linear stability of quasi-periodic tori

Broer

Hoo

Naudot

2007

Journal of Differential Equations

113

View full text Add to dashboard Cite

show abstract

“…This result is based on Theorem C from 1.5. Both results are proved in the forthcoming paper [3]. We have to give an upper bound of the integral not over a vanishing cycle, but over a real oval.…”

Section: Upper Estimates Of Integralsmentioning

confidence: 87%

“…is a connected arc of the path α, and α avoids the β-neighborhoods of the critical values distinct from a j of the polynomial H. Then for any 1-form ω of type (2.2) 4 (3.1) Theorem 3.1 is proved in [3]. It is used in the estimate of the number of zeros in Euclidean disc.…”

Section: Upper Estimates In Euclidean and Poincaré Disksmentioning

confidence: 99%

“…In Section 2 we prove the Main Lemma modulo two statements: formula for the determinant of periods, and upper estimates of Abelian integrals provided by quantitative algebraic geometry. These two statements are treated in two separate papers by the first author (A.Glutsyuk, [2] and [3] respectively). After this the Main Lemma, as well as Theorem A1, is proved.…”

Section: Theorem A2 and Proof Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Restricted Version of the Infinitesimal Hilbert 16th Problem

Glutsyuk

Ilyashenko

2007

MMJ

Self Cite

View full text Add to dashboard Cite

On the number of zeros of Abelian integrals

2010

View full text Add to dashboard Cite

Abstract. We prove that the number of limit cycles generated from nonsingular energy level ovals (periodic trajectories) in a small nonconservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing infinitesimal Hilbert 16th problem.The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection defined over Q (the Gauss-Manin connection) with a quasiunipotent monodromy group.

show abstract

Upper Bounds of Topology of Complex Polynomials in Two Variables

Cited by 5 publications

References 5 publications

Normal linear stability of quasi-periodic tori

Normal linear stability of quasi-periodic tori

Restricted Version of the Infinitesimal Hilbert 16th Problem

On the number of zeros of Abelian integrals

Contact Info

Product

Resources

About