sxy = First[NDSolve[{x'[t]==If[x[t]^2+y[t]^2>2,1,x[t]],y'[t]==If[x[t]^2+y[t]^2>2,-1,y[t]],x[0]==-1/15,y[0]==1/10},{x,y},{t,0,10}]];
pplot=ParametricPlot[Evaluate[{x[t],y[t]} /. sxy],{t,0,10}, PlotStyle->{Thickness[0.01],Red},PlotRange->{{-2,2},{-2,2}},Prolog->{GrayLevel[.9],Disk[{0,0},Sqrt[2]]},AxesLabel->{"x","y"}]

Show[pplot, vf=VectorPlot[If[x^2+y^2>2,{1,-1},{x,y}],{x,-2,2},{y,-2,2},VectorStyle->Gray,VectorPoints->30,VectorScale->{.02,1,None}]]

solutions=Table[First[NDSolve[{x'[t]==If[x[t]^2+y[t]^2>2,1,x[t]],y'[t]==If[x[t]^2+y[t]^2>2,-1,y[t]],x[0]==Cos[θ]/4,y[0]==Sin[θ]/4},{x,y},{t,0,10}]],{θ,0,2π,.1}];
Show[ParametricPlot[Evaluate[{x[t],y[t]}/.solutions],{t,0,10},PlotStyle->Thickness[.01],PlotRange->{{-2,2},{-2,2}},Prolog->{GrayLevel[.9],Disk[{0,0},Sqrt[2]]},AxesLabel->{"x","y"}],vf]

sol3d = First[NDSolve[{x'[t]==Sign[8-y[t]^2-(x[t]+3/2)^2-(z[t]-1)^2],y'[t]==Sign[-y[t]],z'[t]==1,x[0]== 2,y[0]==-2,z[0]==0},{x, y,z},{t,0,5}]];

c1=ContourPlot3D[8-y^2-(x+3/2)^2-(z-1)^2,{x,-2,2},{y,-2,2},{z,0,5},Contours->0,Mesh->False];
c2=ContourPlot3D[-y,{x,-2,2},{y,-2,2},{z,0,5},Contours->0,Mesh->False];

p1=ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.sol3d],{t,0,5},PlotStyle->{Thickness[.015],Red}];
Show[c1,c2,p1,ImageSize->Medium]