چكيده انگليسي :
Computer assisted techniques for the veri cation of the Chebyshev property of Abelian integrals Tahere Mehravar t mehravar@math iut ac ir 2014 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Hamid Reza Zohouri Zangeneh hamidz@cc iut ac ir Advisor Dr Rasoul Asheghi r asheghi@cc iut ac ir 2010 MSC 05C15 53C42 Keywords Hilbert s 16th problem Chebyshev systems Abelian integrals Rigorous computermethods AbstractWe develop techniques for the veri cation of the Chebyshev property of Abelian integrals based onpaper by Birth of canard cycles 8 These techniques are a combination of theoretical results analysisof asymptotic behavior of Wronskians and rigorous computations based on interval arithmetic Weapply this approach to tackle a conjecture formulated by Dumortier and Roussarie in F Dumortier R Roussarie Birth of canard cycles Discrete Contin Dyn Syst 2 2009 723 781 which we areable to prove for q 2 They investigate the number of limit cycles that appear near a slow fast Hopf point i e its cyclicity Their main results show that this cyclicity is nite and modulo the following conjecture provide itsprecise upper bound Conjecture For each integer i 0 let us de ne Ji h y 2i 1 dx where h A x B x y 2 h e 2x x 2 and B x e 2x then J0 J1 Jn is an ECT system on 0 1 for 1 1h with A x 2 2n 0 Figure 1 7 shows Phase portrait of the associated Hamiltonian system with H x y A x B x y 2 To begin with let f0 f1 fn 1 be analytic functions on an interval I De nition 1 f0 f1 fn 1 is a Chebyshev CT system on I if for k 1 2 n any nontrivial
