Traffic information systems make use of the advances in computer and communication technologies to improve performance of the transportation network by providing the traveller with real-time information on travel conditions. Using the information, drivers can en route adjust their choices and make better routing decisions for the remaining part of journey, i.e. make an adaptive route choice. However, instantaneous traffic information becomes obsolete if the majority of drivers switch to other routes in response to the information. This may result in oscillation of traffic flows between alternative routes. The source of this instability lies in the inevitable delay between the time when the traffic conditions at a site are observed and the time when drivers, affected by the information on the traffic conditions, arrive at the site and themselves influence traffic conditions. Up to now, investigation of flow stability under adaptive route choice was performed mainly using simulation experiments. An analytical approach is preferable because it allows general and trustworthy conclusions on qualitative properties of the system. Moreover, it saves extensive time and human resources required for simulation experiments. This paper analytically addresses the stability issue by means of a continuous-time dynamic network model taking into account route updating during a trip. Our specific interest is existence and stability of steady states that correspond to fixed queue lengths and travel times.