Abstract:

This talk deals with a criterion that appears in [M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc. 363  (2011), 109--129], which provides an easy sufficient condition in order for a collection of Abelian integrals $I_0,I_1,ldots,I_{n-1}$ to have the Chebyshev property (i.e., any function in its linear span has at most $n-1$ zeros counted with multiplicities). The condition involves the functions in the integrand of the Abelian integrals and, in the polynomial setting, can be checked in a purely algebraic way by computing resultants and applying Sturm's Theorem.

The first aim of this talk is to show the applicability of above-mentioned criterion in the non-polynomial setting, and to this end we will rely upon computer-assisted methods based on interval arithmetic. In fact we shall study a problem in which the involved functions are non-algebraic. More precisely, we tackle the conjecture posed in [F. Dumortier and R. Roussarie, Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009), 723--781], where the authors consider singular perturbation problems occurring in planar slow-fast systems depending on parameters. They investigate the number of limit cycles that appear near a slow-fast Hopf point, i.e., its cyclicity. Their main results show that under very general conditions this cyclicity is finite and, modulo the following conjecture, provide its precise upper bound.

Conjecture. For each integer $igeq 0,$ let us define $bar J_i(h)=int_{gamma_h}!y^{2i-1}dx,$ where $gamma_hsubset{A(x)+B(x)y^2=h}$ with $A(x)=frac{1}{2}-e^{-2x}(x+frac{1}{2})$ and $B(x)=e^{-2x}.$ Then $bigl(bar J_0,bar J_1,ldots,bar J_nbigr)$ is an extended complete Chebyshev system on $bigr[0,frac{1}{2}bigl)$ for $ngeq 0.$

The second aim of this talk is to show the validity of the conjecture for $n=0,1$ and $2,$ i.e., we prove:

Theorem. $bigl(bar J_0,bar J_1,bar J_2bigr)$ is an ECT-system on $bigr[0,frac{1}{2}bigl).$

The proof is a combination of theoretical results, analysis of asymptotic behaviour of Wronskians and rigorous validation using computer assisted techniques based on interval arithmetic. It is important to point out that, although this theorem does not prove the conjecture for all $n$, it has noteworthy implications. Indeed, one can explicitly bound the cyclicity of a smooth system with a slow-fast Hopf point of codimension 1 or 2, and the cyclicity of an analytic system with a slow-fast Hopf point of order 1 or 2. This is a joint work with Jordi-Llu’s Figueras and Warwick Tucker from Uppsala University (Sweden).