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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...

Smoothness of unordered curves in two-dimensional strongly competitive systems

Janusz Mierczyński (1999)

Applicationes Mathematicae

It is known that in two-dimensional systems of ODEs of the form i = x i f i ( x ) with ∂ f i / ∂ x j < 0 (strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unordered with respect to the coordinatewise (partial) order. We prove that such curves are in fact of class C 1 .

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...

Solution of the 1 : −2 resonant center problem in the quadratic case

Alexandra Fronville, Anton Sadovski, Henryk Żołądek (1998)

Fundamenta Mathematicae

The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field ( x + A 1 x 2 + B 1 x y + C y 2 ) ∂ x + ( - 2 y + D x 2 + A 2 x y + B 2 y 2 ) ∂ y . There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic...

Solutions non oscillantes d’une équation différentielle et corps de Hardy

François Blais, Robert Moussu, Fernando Sanz (2007)

Annales de l’institut Fourier

Soit ϕ : x ↦ ϕ ( x ) , x ≫ 0 une solution à l’infini d’une équation différentielle algébrique d’ordre n , P ( x , y , y ′ , ... , y ( n ) ) = 0 . Nous donnons un critère géométrique pour que les germes à l’infini de ϕ et de la fonction identité sur ℝ appartiennent à un même corps de Hardy. Ce critère repose sur le concept de non oscillation.

Solutions of Riemann–Weber type half-linear differential equation

Ondřej Došlý (2017)

Archivum Mathematicum

We establish an asymptotic formula for a pair of linearly independent solutions of the subcritical Riemann–Weber type half-linear differential equation. We also complement the results of the author and M. Ünal, Acta Math. Hungar. 120 (2008), 147–163, where the equation was considered in the critical case.

Currently displaying 41 – 60 of 197