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Nondegenerate linearizable centre of complex planar quadratic and symmetric cubic systems in C2.

Colin Christopher, Christiane Rousseau (2001)

Publicacions Matemàtiques

In this paper we consider complex differential systems in the plane, which are linearizable in the neighborhood of a nondegenerate centre. We find necessary and sufficient conditions for linearizability for the class of complex quadratic systems and for the class of complex cubic systems symmetric with respect to a centre. The sufficiency of these conditions is shown by exhibiting explicitly a linearizing change of coordinates, either of Darboux type or a generalization of it.

On gradient at infinity of semialgebraic functions

Didier D'Acunto, Vincent Grandjean (2005)

Annales Polonici Mathematici

Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number ϱ c ≤ 1 such that |x|·|∇f| and | f ( x ) - c | ϱ c are separated at infinity. If c is a regular value and ϱ c < 1 , then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.

On the existence of canard solutions

Daniel Panazzolo (2000)

Publicacions Matemàtiques

We study the existence of global canard surfaces for a wide class of real singular perturbation problems. These surfaces define families of solutions which remain near the slow curve as the singular parameter goes to zero.

On the existence of chaotic behaviour of diffeomorphisms

Michal Fečkan (1993)

Applications of Mathematics

For several specific mappings we show their chaotic behaviour by detecting the existence of their transversal homoclinic points. Our approach has an analytical feature based on the method of Lyapunov-Schmidt.

On the number of zeros of Melnikov functions

Sergey Benditkis, Dmitry Novikov (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

We provide an effective uniform upper bound for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order k of the Melnikov function. The generic case k = 1 was considered by Binyamini, Novikov and Yakovenko [BNY10]. The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.

Pseudo-abelian integrals on slow-fast Darboux systems

Marcin Bobieński, Pavao Mardešić, Dmitry Novikov (2013)

Annales de l’institut Fourier

We study pseudo-abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.

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