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Moving singularities of a cyclic system of differential equations. (Russian) Zbl 0738.34006

Let \(R\) be a ring of germs of functions holomorphic at the origin and \(\omega\) be the usual order function on \(R\). The authors discuss a system of differential equations of the form (1) \(dW_ j/dz=\sum_{k=0}^{m_ j}p_{kj}W_{j+1}^{m_ j-k}\), where \(j=1,\dots,n\), \(W_{n+1}=W_ 1\), \(m_ j\in\mathbb{Z}^ +\), \(p_{kj}\in R\), \(\prod_{j=1}^ n m_ j>1\) and \(\omega(p_{0i})=0\) for some \(i\). They prove (Theorem 1) that any formal solution of system (1) is convergent, and give the formulae for computing \(\omega(W_ j)\) \((j=1,\dots,n)\) for such a solution. In the paper a parametrization for the families of these solutions is obtained (Theorems 2,3) in the cases where \(n=3,4\).

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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