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　(q)　is　equal　to　or　less　than　0.923.　The　bifurcation　appears　in　the　system　when　q　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　q　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.　©　2018　by　the　authors.