The integrability of strongly degenerate singularity for a class of planar analytical systems
The integrability problems of strong degenerate singularities for a type of planar analytical systems are studied by means of the normal form theory.Firstly,it is proved that the integrability of any differentiable system remains unchanged under the variable transformation of the differentiable homeomorphism and time scale transformation.Secondly,the quadratic homogeneous differential system can be transformed into four canonical forms by affine transformation and time scale transformation.Finally,as for one of the canonical forms,the conditions of the integrability for the corresponding nonlinear analytical systems are given by using the normal form theory together with the necessary and sufficient conditions that the main system is Hamiltonian system and that the factorization of the corresponding Hamiltonian function only has simple factors.The results provide a theoretical basis for studying the phase diagram and local qualitative structure of planar systems.
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