Study On Monodromy Problem Of Strongly Degenerate Singular Points For A Class Of Planar Systems

The monodromy problem is a previous step to solve the center problem of planar differential systems which is one of the open classical problems in the qualitative theory of planar differential systems.If the linearized matrix of the system at the singular point is not identically zero,the monodromy problem has been completely solved.If the linearized matrix of the system at the singular point is identically zero,there are few results,the monodromy problem has not been completely solved at present.The first work of this dissertation is to study the invariance of monodromic orbits under which coordinate transformation of planar analytic systems.Firstly,two definitions of orbits of the planar analytic system entering a singular point along a fixed direction are given,and they are proved to be equivalent.Secondly,a sufficient and necessary condition for discriminating that orbits enter a singular point along a fixed direction and the examples of singular point of planar analytical system with different monodromy under non-regular coordinate transformation are introduced.Finally,it is proved that the monodromy of singular point of the planar analytic system remains invariant under regular coordinate transformation,and two corollaries are given for a given planar linear homogeneous constant coefficient system,the invariance of its monodromic orbit can be proved by using the obtained results.The second work of this dissertation is to study the monodromy of subcubic differential systems(that is,differential systems whose lowest degree is cubic).Firstly,the Newton diagram definition and the conservative-dissipative decomposition method of the vector field corresponding to the planar differential system are given,the bounded edges of the Newton diagram of subcubic differential systems are used for homogeneous decomposition(special quasihomogeneous decomposition),it is expressed as the sum of the conservative term(having zerodivergence)and the dissipative term(in the sense of non-conservative part that fully captures the divergence of vector field).Secondly,the type and degree of the system are obtained according to the definition of the vector field of quasi-homogeneous polynomial,and the Hamilton function of the lowest-degree quasi-homogeneous term of the subcubic differential system is obtained by using the quasi-homogeneous decomposition formula,which is expressed in compact form.Then,for a given subcubic differential system,the monodromy of the system is determined based on the conditions satisfied by its Newton diagram and the Hamilton function of the lowest-degree quasi-homogeneous term.Finally,it is proved that the singular points of eight of the ten classes of standard forms of subcubic differential systems are non-monodromic,and that of two systems are monodromic.