f[x_,v_]:=v;
g[x_,v_]:=(1-3*x^2-2*v^2)*v-x;

plot1=ContourPlot[f[x,v]==0,{x,-1.5,1.5},{v,-1.5,1.5},FrameLabel->{"x","v"},FrameStyle->Directive[Black,Large],PlotRange->{{-1.5,1.5},{-1.5,1.5}},ContourStyle->{Thick,Red},PlotPoints->100];
plot2=ContourPlot[g[x,v]==0,{x,-1.5,1.5},{v,-1.5,1.5},ContourStyle->{Thick,Blue},PlotPoints->100];
plot3=StreamPlot[{f[x,v],g[x,v]},{x,-1.5,1.5},{v,-1.5,1.5}];
s=NDSolve[{x'[t]==f[x[t],v[t]],v'[t]==g[x[t],v[t]],x[0]==.75,v[0]==0},{x,v},{t,30}];
plot4=ParametricPlot[Evaluate[{x[t],v[t]}/. s],{t,0,30},PlotStyle->{Thick,Black}];
Show[plot1,plot2,plot3,plot4]