to horizontal frictionless plane has features of simple dynamic differential equations and separable variables. In

this study polar coordinates are used to construct the dynamic differential equations of the spring-mass oscillator

confined to horizontal frictionless plain. Variables related to radial motion are separated to form an individual dynamic

differential equation，which is researched by qualitative analysis and approximation theory due to its

nonlinear characteristic. Software MATLAB is used as a numerical method to simulate paths of the oscillator，verify

qualitative results and demonstrate the accuracy of approximate solutions in two specific situations.