This　study　aimed　to　reveal　the　evolution　rule　of　day-to-day　traffic　flow　with　a　variable　quantity　of　shared　parking　lots.　To　achieve　this,　a　trading　system　for　shared　parking　lots　was　developed.　Shared　parking　providers　can　choose　between　rational　and　naive　expectations　of　the　prices　of　shared　parking　lots　in　the　trading　system.　On　this　basis,　we　investigated　the　evolution　rule　of　the　equilibrium　price　of　shared　parking　lots,　proportion　of　two　types　of　parking　providers,　and　travel　demands　of　public　transit　and　private　vehicles　during　rush　hour.　Subsequently,　taking　a　network　with　two　parallel　roadways　as　an　example,　we　examined　the　final　state　of　the　dynamical　evolution　of　the　network　flow　system.　A　quantitative　criterion　is　provided　for　the　final　evolution　state　of　both　systems.　A　numerical　experiment　with　a　real　traffic　network　in　Beijing　was　conducted.　Theoretical　analysis　and　numerical　results　lead　to　the　following　conclusions:　(1)　Concerning　the　trading　system　of　shared　parking　lots,　if　the　slope　of　the　supply　curve　is　less　than　that　of　the　demand　curve,　the　unique　equilibrium　solution　of　the　trading　system　of　shared　parking　lots　can　always　achieve　an　asymptotically　stable　state.　If　the　summation　of　transaction　and　forecasting　costs　for　rational　providers　is　higher　than　that　for　naive　providers,　a　critical　choice　intensity　of　parking　providers　exists,　which　may　lead　to　bifurcation　and　chaos　in　the　trading　system.　(2)　For　a　network　flow　system,　the　unique　equilibrium　solution　may　be　asymptotically　stable　if　the　travel　cost　is　not　sensitive　to　the　path　flow.　The　selected　path　probability　is　not　sensitive　to　the　travel　cost　if　the　vehicle　demand　is　not　large;　otherwise,　the　system　can　incur　bifurcation　and　chaos.　(3)　When　the　trading　system　reaches　an　asymptotically　stable　state,　if　parking　providers　(users)　are　insensitive　(sensitive)　to　parking　prices　while　the　potential　parking　demand　is　low,　the　summation　of　transaction　and　forecasting　costs　for　the　rational　provider　is　not　much　higher　than　that　for　the　naive　provider,　and　the　choice　intensity　of　parking　providers　is　not　large.　The　equilibrium　demand　for　private　vehicles　during　rush　hour　is　not　high　because　of　the　limited　number　of　reserved　shared　parking　lots,　which　leads　to　a　final　evolutionary　state　of　the　network　flow　system　tending　to　be　asymptotically　stable.　(4)　Once　chaos　occurs　in　the　trading　system　of　shared　parking　lots,　the　network　flow　system　may　generate　more　serious　bifurcation　and　chaos.　©　2022　Xi＇an　Highway　University.　All　rights　reserved.