## 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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### References:

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