About Integrability of the Degenerate System
We study integrability of an autonomous planar polynomial system of ODEs with a degenerate singular point at the origin depending on five parameters. By mean of the Power Geometry Method, this degenerated system is reduced to a non-degenerate form by the blow-up process. After, we search for the necessary conditions of local integrability by the normal form method. We look for the set of necessary conditions on parameters under which the original system is locally integrable near the degenerate stationary point. We found seven two-parametric families in the five-parameter space. Then first integrals of motion were found for six families. For the seventh family, we found the formal first integral. So, at least six of these families in parameters space are manifolds where the global integrability of the original system takes place.
KeywordsOrdinary differential equations Integrability Resonant normal form Power geometry Computer algebra
The author is very grateful to Profs. A.D. Bruno, V.G. Romanovski, and A.B. Batkhin for important advices, discussions, and assistance.
- 3.Bruno, A.D.: Analytical form of differential equations (I, II). Trudy Moskov. Mat. Obsc. 25, 119–262 (1971), 26, 199–239 (1972) (in Russian). Trans. Moscow Math. Soc. 25, 131–288 (1971), 26, 199–239 (1972) (in English)Google Scholar
- 4.Bruno, A.D.: Local Methods in Nonlinear Differential Equations. Nauka, Moscow 1979 (in Russian). Springer, Berlin (1989) (in English)Google Scholar
- 5.Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. Fizmatlit, Moscow (1998) (in Russian). Elsevier Science, Amsterdam (2000) (in English)Google Scholar
- 7.Edneral, V., Romanovski, V.G.: On sufficient conditions for integrability of a planar system of odes near a degenerate stationary point. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2010. LNCS, vol. 6244, pp. 97–105. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15274-0_9CrossRefGoogle Scholar
- 11.Bruno, A.D., Edneral, V.F.: On possibility of additional solutions of the degenerate system near double degeneration at the special value of the parameter. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 75–87. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-02297-0_6CrossRefGoogle Scholar
- 14.Edneral, V., Romanovski, V.: Local and global odes properties. In: Proceedings of 24th Conference on Applications of Computer Algebra - ACA, Santiago de Compostela, Spain (2018). https://doi.org/10.15304/9788416954872