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Successive derivatives of a first return map, application to the study of quadratic vector fields. (English) Zbl 0852.34008

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 \(H= 1/2(x^2+ y^2)\).

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
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