The dynamic behavior of thermodynamic systems, described by one order parameter and one control parameter [1], as well as by one order parameter and several control parameters were studied in a small neighborhood of ordinary and bifurcation equilibrium values of the system parameters. The analysis of order parameter dependences on the control parameter(s) in a small vicinity of the equilibrium values of parameters, including the stability and sensitivity analyses of the equilibrium states, and the asymptotic behavior of the order parameter dependences on the control parameter (bifurcation diagrams) were done by using the general methods of investigating the branching (bifurcations) of solutions for nonlinear equations.  One can estimate the transition time to the unstable state in the neighborhood of ordinary and bifurcation equilibrium values of parameters. The influence of an external field on the dynamic behavior of thermodynamic system is also analyzed, and the peculiarities of the system dynamic behavior are discussed near the ordinary and bifurcation equilibrium values of parameters in the presence of external field. It makes possible to obtain a general classification of the dependences in the small neighborhood of steady states. We note that such representation was used partially in [2] when constructing the formalism of thermodynamic geometry. In context of our exhaustive analysis of the dynamic behavior of thermodynamic systems described by one order parameter and one (or several) control parameter(s) in a small neighborhood of the equilibrium states of these systems, we prove the existence of five canonical (normal) forms of bifurcations and a much larger number of bifurcation diagrams than it was reported in the Ref. [2]. The expressions for canonical forms are presented through the expressions for the coefficients of these forms, that is, the information about the thermodynamic system is preserved, which is very important in the study of the stability of equilibrium states. The dynamic process of magnetization of a ferromagnet was discussed by using the general methods of bifurcation and stability analysis [1]. One can note that the Landau-type kinetic potential was used in the theoretical analysis and calculations of such ―nonclassical‖ nucleation pathways. e.g. the experimental observations on anomalous generation and extinction phenomenon of crystal nuclei at very low temperatures in non-equilibrium supercooled liquids containing hydroxyl group, namely o-benzylphenol [3], salol [4], and 2,2'- dihydroxybenzophenone [5], observed during the progress of crystal nucleation and growth below the glass transition temperature. We shall also mention that the general analysis of equilibrium states, their bifurcations, and stability for systems described by one order parameter and several control parameters is presented in the Ref. [6], and a global analysis of the equilibrium states of the system, as well as their bifurcation and stability analysis, was discussed in detail in Ref. [6, 7].