This paper proposes and investigates a new equilibrium equivalence concept, which is more general than the conventional topological equivalence concept but offers easier analysis and better engineering insights for nonlinear system design. We prove an equilibrium Equivalence Theorem which guarantees the robustness of the system dynamics under structured perturbations with certain fairly general assumptions. Concepts of Equilibrium Equivalence and Equilibrium Equivalence Structural Stability are developed and applied to studies of bifurcations of vector fields on noncompact manifolds. Constructive approach to Equilibrium Equivalence Structural Stability verification is emphasized. Some applications to nonlinear stability analysis are demonstrated.