Francoise, J. P. Successive derivatives of a first return map, application to the study of quadratic vector fields. (English) Zbl 0852.34008 Ergodic Theory Dyn. Syst. 16, No. 1, 87-96 (1996). A two-dimensional system of differential equations \[ \dot x= {\partial H\over \partial y}+ \varepsilon f,\quad \dot y= - {\partial H\over \partial x}+ \varepsilon g, \] where \(H(x, y)\), \(f(x, y)\), \(g(x, y)\) are polynomials, and the level lines \(H= r\) are compact for small \(r\geq 0\), is considered. The author investigates a return map \(L: r\to L(r, \varepsilon)\) under a certain additional condition on \(H\) which is satisfied for \(H= 1/2(x^2+ y^2)\). An algorithm to compute for any \(k\) the derivative \({\partial^k L\over \partial \varepsilon^k} (r, 0)\) which is not identically zero, is given. It is shown how this algorithm works in the case \(H= 1/2(x^2+ y^2)\). Reviewer: Yu.N.Bibikov (St.Peterburg) Cited in 8 ReviewsCited in 100 Documents MSC: 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion Keywords:two-dimensional system of differential equations; return map; derivative PDFBibTeX XMLCite \textit{J. P. Francoise}, Ergodic Theory Dyn. Syst. 16, No. 1, 87--96 (1996; Zbl 0852.34008) Full Text: DOI References: [1] Devlin, Math. Proc. Camb. Phil. Soc. 110 pp 569– (1991) [2] DOI: 10.2307/2000999 · Zbl 0678.58027 · doi:10.2307/2000999 [3] Bao, Preprint (1993) [4] Bautin, Amer. Math. Soc. Transl. 5 pp 396– (1962) [5] Dulac, Bull. Sci. Math. 32 pp 230– (1908) [6] DOI: 10.1006/jdeq.1994.1049 · Zbl 0797.34044 · doi:10.1006/jdeq.1994.1049 [7] Pugh, Springer Lecture Notes in Mathematics 468 pp 55– (1974) [8] DOI: 10.1007/BF01390315 · Zbl 0448.34012 · doi:10.1007/BF01390315 [9] DOI: 10.1016/0022-0396(86)90030-6 · Zbl 0602.34019 · doi:10.1016/0022-0396(86)90030-6 [10] Yakovenko, Preprint (1993) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.