System:x'=k x,y'=m y

x’=-x,y’=-y

sys1aplot=StreamPlot[{-x,-y},{x,-5,5},{y,-5,5}]

sys1alines=Plot[{0,x,2x,-x,-2x,x/2,-x/2,100x},{x,-5,5},PlotStyle->{{Thick, Black}},PlotRange->{-5,5}];
Show[sys1aplot,sys1alines]

The critical point (0,0) is a stable node or sink.

x’=-x,y’=-2y

sys1bplot=StreamPlot[{-x,-2y},{x,-5,5},{y,-5,5}]

sys1blines=Plot[{0,x^2,-x^2,x^2/5,-x^2/5,100x},{x,-5,5},PlotStyle->{{Thick, Black}},PlotRange->{-5,5}];
Show[sys1bplot,sys1blines]

The critical point (0,0) is also a stable node.
Both are actually asymptotically stable: solutions close enough to (0,0) converge to it.

x’=x,y’=y

sys1cplot=StreamPlot[{x,y},{x,-5,5},{y,-5,5}]

Show[sys1cplot,sys1alines]

The critical point (0,0) is an unstable node or source.

x’=x,y’=3y

sys1dplot=StreamPlot[{x,3y},{x,-5,5},{y,-5,5}]

sys1dlines=Plot[{0,(x^3)/4,-(x^3)/4,(x^3)/25,-(x^3)/25,100x},{x,-5,5},PlotStyle->{{Thick, Black}},PlotRange->{-5,5}];Show[sys1dplot,sys1dlines]

The critical point (0,0) is also an unstable node.

x’=x,y’=-y

sys1eplot=StreamPlot[{x,-y},{x,-5,5},{y,-5,5},StreamPoints->Automatic]

sys1elines=Plot[{0,1/x,-1/x,3/x,-3/x,7/x,-7/x,11/x,-11/x,100x},{x,-5,5},PlotStyle->{{Thick, Black}},PlotRange->{-5,5}];
Show[sys1eplot,sys1elines]

The critical point (0,0) is a saddle point.

System: x'=2y+x-1,y'=(x^2)-2x-3

sys2xnull=ContourPlot[{x==-1,x==3},{x,-5,5},{y,-5,5},ContourStyle->Blue];
sys2ynull=ContourPlot[{2y+x-1==0},{x,-5,5},{y,-5,5},ContourStyle->Red];
sys2plot=StreamPlot[{2y + x-1,(x^2)-2x - 3},{x,-5,5},{y,-5,5},StreamPoints->Fine];

Show[sys2xnull,sys2ynull]

Show[sys2xnull,sys2ynull,sys2plot]

Critical points:unstable spiral point at (-1,1),saddle point at (3,-1)

System: x'=4y,y'=-x

sys3plot=StreamPlot[{4y, -x},{x,-5,5},{y,-5,5}]

sys3contour[c_]=(x^2)+4(y^2)==c;sys3lines=ContourPlot[Evaluate[Map[sys3contour,{1,9,25,49,81}]],{x,-5,5},{y,-5,5},ContourStyle->{{Thick,Black}}];
Show[sys3plot,sys3lines]

The critical point (0,0) is a center.
It is stable, but not asymptotically stable: solutions nearby (0,0) stay close, but do not converge to it.

System: x'=-y-x (((1-(x^2)-(y^2)))^2),y'=x-((y(1-(x^2)-(y^2)))^2)

sys4plot=StreamPlot[{-y-x(((1-((x^2)+(y^2))))^2), x-y(((1-((x^2)+(y^2))))^2)},{x,-2,2},{y,-2,2}]

sys4lines=ContourPlot[(x^2)+(y^2)==1,{x,-2,2},{y,-2,2},ContourStyle->{{Thick,Black}}];
Show[sys4plot,sys4lines]

Limit cycle on curve (x^2)+(y^2)=1; asymptotically stable spiral point at (0,0).