This paper aims to study the conditions of traffic flow evolving to stability and the stability of equilibrium under demand time-varying of traffic networks． A general framework of the evolution of flow dynamics by adopting evolutionary game theory and dynamic system stability theory is proposed． A multi-group and multicriteria travel choice flow evolution dynamic model is given． The equivalence of the balance position of flow evolution dynamic model and equilibrium solutions of dynamic traffic assignment model is proved．Furthermore，the existence，uniqueness and stability of the solutions are discussed． Theoretical analysis results show that unique solutions of evolutionary dynamic model always exist in a local region． Different stability properties will emerge nearby the equilibrium solutions of flow evolution dynamic model under the hypothesis of traveler individual income parameters satisfying certain conditions． Rationality and validity of the proposed model are addressed through a simple numerical example． Links between travelers’game behavior and dynamic traffic assignment are established． These conclusions deepen our understanding of urban traffic flow evolution law．