Normal Forms of the Degenerate Autonomous Differential Equations with the Maximal Jordan Chain and Simple Applications

Degenerate differential equations, as part of the differential-algebraic equations, the last few decades cause increasing interest among researchers, both because of the attractiveness of the considered theoretical questions, and by virtue of their applications. Currently, advanced methods developed in this area are used for system modelling and analysis of electrical and electronic circuits, chemical reaction simulations, optimization theory and automatic control, and many other areas. In this paper, the theory of normal forms of differential equations, originated in the works of Poincare and recently developed in the works of Arnold and his school, adapted to the simplest case of a degenerate differential equations. For this purpose we are using technique of Jordan chains, which was widely used in various problems of bifurcation theory. We study the normal forms of degenerate differential equations in the case of the existence of the maximal Jordan chain. Two and three dimensional spaces are studied in detail. Normal forms are the simplest representatives of the degenerate differential equations, which are equivalent to more complex ones. Therefore, normal forms should be considered as a model type of degenerate differential equations.

Keywords

degenerate differential equations; normal forms; Jordan chains.

References

1. Vainberg M.M., Trenogin V.A. Theory of Branching of Solutions of Non-Linear Equations. Leyden, Nordhoof International Publishing, 1974.
2. Arnold V.I. Geometricheskie metody v teoriy obyknovennykh differentsialnykh uravneniy [Geometrical Methods in the Theory of Ordinary Differential Equations]. Мoscow, Moscow Center for Continuous Mathematical Education, 1999.
3. Shui-Nee Chow, Chengzhi Li, Duo Wang. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, 1994.
4. Jooss G., Adelmeyer M. Topics in Bifurcation Theory and Applications. Singapore, New Jersey, London, Hong Kong, World Scientific, 1992.
5. Loginov B.V., Rousak Yu.B., Kim-Tyan L.R. Normal Forms of the Degenerate Differential Autonomous and Non-Autonomous Equations with the Maximal Jordan Chain of Length Two and Three. The Bulletin of Irkutsk State University. Series: Mathematics, 2015, vol. 12, pp. 58-71. (in Russian)
6. Loginov B.V., Rousak Yu.B., Kim-Tyan L.R. Normal Forms for the Degenerate Non-Autonomous Differential Equations in the Spaces R^{n}, n=2,3,4. Sbornik nauchnykh trudov 'Prikladnaya matematika i mekhanika', Ulyanovsk, 2014, no. 10, pp. 142-160. (in Russian)
7. Loginov B.V., Rousak Yu.B., Kim-Tyan L.R. Differential Equations with Degenerated Variable Operator at the Derivative. Current Trends in Analysis and Its Applications. Proceedings of the 9th ISAAC Congress, Krakow 2013, 2015, pp. 101-108. DOI: 10.1007/978-3-319-12577-0_14
8. Marszalek W. Fold Points and Singularity Induced Bifurcation in Inviscid Transonic Flow. Physics Letters A, 2012, vol. 376, issues 28-29, pp. 2032-2037. DOI:10.1016/j.physleta.2012.05.003
9. Stepanov V.V. Kurs differentsialnykh uravneniy [The Course of the Differential Equations]. Moscow, Gosudarstvennoe izdatel'stvo tekhniko-teoreticheskoy literatury, 1950.