This paper concentrates on traveling wave approximation, phase plane, and stability analysis of non-Newtonian fluids. The analysis is extended for the dynamical system theory of the problem to understand the flow behavior. Higher-order nonlinear autonomous differential equations are studied. These equations characterize the trajectories of the particles. The phase plots are drawn to show the qualitative behavior of the fluid flow. Equilibrium points are calculated and bifurcation diagrams are drawn for the complete range of parameters. Equilibria plots depict a detailed discussion of various flow patterns developed for the complete range of flow variables contrary to the current which describes the topological structure at some particular value of the parameter. Also, compare our numerical solution to the HAM solution and get an exceptional result.