Stochastic　noise　is　common　in　many　engineering　systems.　Stochastic　noise　and　nonlinearity　of　dynamical　systems　will　lead　to　the　occurrence　of　complex　dynamic　phenomena,　such　as　stochastic　bifurcations　and　transitions.　In　this　paper,　a　radial　basis　function　neural　networks　(RBF-NN)　method　is　applied　to　study　stochastic　bifurcations　and　transient　dynamics　of　probability　response.　A　Duffing　oscillator　under　additive　and　multiplicative　random　noise　is　presented　to　show　the　effectiveness　of　the　proposed　solution　method.　A　new　type　of　stochastic　bifurcations　is　found　with　an　increase　of　system　control　parameter,　which　is　called　a　stochastic　double　P-bifurcation.　It　occurs　when　a　stationary　probability　density　function　(PDF)　changes　from　a　single　peak　to　another　single　one　through　two　double　peaks　with　their　maximum　exchange.　It　is　revealed　that　such　a　double　P-bifurcation　is　linked　to　two　catastrophic　bifurcations　of　its　deterministic　counterpart.　Moreover,　these　different　types　of　transient　evolutions　of　the　PDFs　are　observed　with　transitions　of　probability　maximum　peak　to　one,　and　to　the　other,　as　well　as　to　the　both　of　stable　invariant　sets　from　different　initial　PDFs.　The　evolutionary　orientation　of　transient　PDFs　aligns　with　unstable　invariant　manifolds　leading　to　stable　invariant　sets.　The　previous　conjecture　is　confirmed　that　the　noise-induced　transient　dynamics　and　bifurcations　are　dominated　by　the　global　topology　and　its　sudden　changes　of　corresponding　deterministic　systems　without　noise.　©　2022　Elsevier　Ltd