EuDML | Browse

We show that for a generic polynomial $H = H (x, y)$ and an arbitrary differential 1-form $ω = P (x, y) d x + Q (x, y) d y$ with polynomial coefficients of degree $\leq d$ , the number of ovals of the foliation $H = const$ , which yield the zero value of the complete Abelian integral $I (t) = \oint_{H = t} ω$ , grows at most as $exp O_{H} (d)$ as $d \to \infty$ , where $O_{H} (d)$ depends only on $H$ . The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let $f_{1} (t), \dots, f_{n} (t)$ , $t \in K ⋐ ℝ$ , be a fundamental system of real solutions...

Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks

Masafumi Yoshino, Todor Gramchev (2008)

Annales de l’institut Fourier

We study the simultaneous linearizability of $d$ –actions (and the corresponding $d$ -dimensional Lie algebras) defined by commuting singular vector fields in $ℂ^{n}$ fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of $d$ vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then...

Singularities of Vector Fields on the Plane With Pointed Direction.

R.I. Bogdanov (1979)

Inventiones mathematicae

Small amplitude critical periods in a cubic polynomial system.

Toni, Bourama (1999)

Southwest Journal of Pure and Applied Mathematics [electronic only]

Smooth Gevrey normal forms of vector fields near a fixed point

Laurent Stolovitch (2013)

Annales de l’institut Fourier

We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $α$ -Gevrey vector field with an hyperbolic linear part admits a smooth $β$ -Gevrey transformation to a smooth $β$ -Gevrey normal form. The Gevrey order $β$ depends on...

Smooth invariant curves of singularities of vector fields on $ℝ^{3}$

Patrick Bonckaert (1986)

Annales de l'I.H.P. Analyse non linéaire

Smooth linearization of centres

Massimo Villarini (2000)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Smooth linearization of germs of $R^{2}$ -actions and holomorphic vector fields

F. Dumortier, Robert Roussarie (1980)

Annales de l'institut Fourier

The paper contains a generic condition permitting the linearization in class $𝒞^{k}$ , $0 \leq k \leq \infty$ , of germs of singular infinitesimal $R^{2}$ -actions on $R^{n}$ $(n \geq 2)$ and of singular holomorphic...

Smooth normalization of a vector field near a semistable limit cycle

Sergey Yu. Yakovenko (1993)

Annales de l'institut Fourier

We establish a polynomial normal form for a vector field having a limit cycle of multiplicity 2. The smooth classification problem for such fields is closely related to the problem of classification of germs $Δ : (ℝ^{1}, 0) \to (ℝ^{1}, 0)$ , $Δ (x) = x + c x^{2} + \dots$ , solved by F. Takens in 1973. Such germs appear as the germs of Poincaré return maps for semistable cycles, and a smooth conjugacy between any two such germs may be extended to a smooth orbital equivalence between the original fields.If one deals with smooth conjugacy of flows rather than...

Smooth transformations of intervals

Oscar E. Lanford III (1980/1981)

Séminaire Bourbaki

Smoothness property for bifurcation diagrams.

Robert Roussarie (1997)

Publicacions Matemàtiques

Strata of bifurcation sets related to the nature of the singular points or to connections between hyperbolic saddles in smooth families of planar vector fields, are smoothly equivalent to subanalytic sets. But it is no longer true when the bifurcation is related to transition near singular points, for instance for a line of double limit cycles in a generic 2-parameter family at its end point which is a codimension 2 saddle connection bifurcation point. This line has a flat contact with the line...

We discuss the existence of closed geodesic on a Riemannian manifold and the existence of periodic solution of second order Hamiltonian systems.

Currently displaying 1 – 20 of 32

Page 1 Next

Displaying 1 – 20 of 32

Saddle-focus singular cycles and prevalence of hyperbolicity

Second order linear differential systems

Separatrix conditions yielding either periodic orbits or unusual behavior for flows on M3.

Separatrix conditions yielding either periodic orbits or unusual behavior for flows on M3. (Short Communication).

Shift of bifurcation point due to noise induced parameter.

Simple exponential estimate for the number of real zeros of complete abelian integrals