This paper proposes a traffic-flow evolutionary model under a dual updating mechanism that describes the day-to-day (DTD) dynamics of traffic flow and travel cost. To illustrate the concept, a simple two-route network is considered. Based on the nonlinear dynamic theory, the equilibrium stability condition of the system is derived and the condition for the division between the bifurcation and chaotic states of the system is determined. The characteristics of the DTD dynamic evolution of network traffic flow are investigated using numerical experiments. The results show that the system is absolutely stable when the sensitivity of travelers toward the route cost parameter (θ) is equal to or less than 0.923. The bifurcation appears in the system when θ is larger than 0.923. For values of θ equal to or larger than 4.402, the chaos appears in the evolution of the system. The results also show that with the appearance of chaos, the boundary and interior crises begin to appear in the system when θ is larger than 6.773 and 10.403, respectively. The evolution of network traffic flow is always stable when the proportion of travelers who do not change the route is 84% or greater.