A　Kerr　nonlinear　oscillator　(KNO)　supports　a　pair　of　steady　eigenstates,　coherent　states　with　opposite　phases,　that　are　good　for　the　encoding　of　continuous　variable　qubit　basis　states.　Arbitrary　control　of　the　KNO　confined　within　the　steady-state　subspace　allows　extraction　of　the　Berry　curvature　through　the　linear　response　of　the　physical　observable　to　the　quench　velocity　of　the　system,　providing　an　effective　method　for　the　characterization　of　topology　in　the　KNO.　As　an　alternative,　the　control　adopting　the　＂shortcut　to　adiabaticity＂　to　the　KNO　enables　the　exploration　of　the　topology　through　accelerated　adiabatic　eigenstate　evolution　to　measure　all　3　physical　observables.　Topological　transitions　are　revealed　by　the　jump　of　the　first　Chern　number,　obtained　respectively　from　the　integral　of　the　Berry　curvature　and　of　the　new　polar　angle　relation,　over　the　whole　parameter　space.　Our　strategy　paves　the　way　for　measuring　topological　transitions　in　continuous　variable　systems.