Network　traffic　flow　is　an　aggregated　result　of　a　huge　number　of　travelers＇　route　choices,　which　is　influenced　by　the　travelers＇　choice　behaviors.　So　day-to-day　traffic　flow　is　not　static,　but　presents　a　complex　and　tortuous　day-to-day　dynamic　evolution　process.　Studying　day-to-day　dynamic　evolution　of　network　traffic　flow,　we　can　not　only　know　whether　the　traffic　network　equilibrium　can　be　reached　and　how　the　process　is　achieved,　but　also　can　know　what　phenomenon　will　occur　in　the　evolution　of　network　traffic　flow　if　the　equilibrium　is　not　reached.　In　a　real　traffic　system,　taking　day　as　scale　unit,　the　day-to-day　network　traffic　demand　is　variable　and　changes　with　everyday＇s　traffic　network　state.　The　travelers＇　route　choices　are　also　influenced　by　the　previous　day＇s　behaviors　and　network　state.　Then,　will　the　day-to-day　network　traffic　flow　evolution　be　stable?　If　it　is　unstable,　when　will　bifurcation　and　chaos　occur?　In　this　paper　we　discuss　the　day-to-day　dynamic　evolution　of　network　traffic　flow　with　elastic　demand　in　a　simple　two-route　network.　The　dynamic　evolution　model　of　network　traffic　flow　with　elastic　demand　is　formulated.　Based　on　a　nonlinear　dynamic　theory,　the　existence　and　uniqueness　of　the　fixed　point　of　dynamic　evolution　model　are　proved,　and　an　equilibrium　stability　condition　for　the　dynamic　evolution　of　network　traffic　flow　with　elastic　demand　is　derived.　Then,　the　evolution　of　network　traffic　flow　is　investigated　through　numerical　experiments　by　changing　the　three　parameters　associated　with　travelers,　which　are　the　sensitivity　of　travelers＇　travel　demand　to　travel　cost,　the　randomness　of　travelers＇　route　choices,　and　travelers＇　reliance　on　the　previous　day＇s　actual　cost.　Our　findings　are　as　follows.　Firstly,　there　are　three　kinds　of　final　states　in　the　evolution　of　network　traffic　flow:　stability　and　convergence　to　equilibrium,　periodic　motion　and　chaos.　The　final　state　of　the　network　traffic　flow　evolution　is　related　to　the　above　three　parameters.　It　is　found　that　under　certain　conditions　the　bifurcation　diagram　of　the　network　traffic　flow　evolution　reveals　a　complicated　phenomenon　of　period　doubling　bifurcation　to　chaos,　and　then　period-halving　bifurcation.　Meanwhile,　the　chaotic　region　is　interspersed　with　odd　periodic　windows.　Moreover,　the　more　sensitive　to　cost　the　travelers＇　travel　demand　the　more　likely　the　system　evolution　is　to　be　stable.　The　smaller　the　randomness　of　travelers＇　route　choices,　the　less　likely　the　system　evolution　is　to　be　stable.　The　lower　the　degree　of　travelers＇　reliance　on　the　previous　day＇s　actual　cost,　the　more　likely　the　system　evolution　is　to　be　stable.　©　2017　Chinese　Physical　Society.