Exercises 3.1.a

For:

x
=2+rx+
2
x
i) Sketch the different vector field types that appear when you vary
r
. We’re actually going to do this first the hard way, then the easier way.
​
Looking at the phase portrait for different values of
r
we get the following.
​
In[]:=
finsol[r_]:=Select[(x/.NSolve[2+rx+
2
x
0,x]),Head[#]Real&];GraphicsGrid[Partition[Show[Plot[2+#x+
2
x
,{x,-5,6},AxesLabelEvaluate[Style[#,14]&/@{"x","

x
"}],PlotLabel"r = "<>ToString[#]],If[finsol[#]==={},ListPlot[{{10,10}}],If[Min[finsol[#]]>10,Show[{Graphics[Circle[{finsol[#][[1]],0},0.2]],Graphics[Disk[{finsol[#][[2]],0},0.2]]}],Show[{Graphics[Disk[{finsol[#][[1]],0},0.2]],Graphics[Circle[{finsol[#][[2]],0},0.2]]}]]],If[finsol[#]==={},Graphics[{Thick,Arrow[{{-5,0},{6,0}}]}],{Graphics[{Thick,Arrow[{{-5,0},{Min[finsol[#]]-0.2,0}}],Arrow[{{Max[finsol[#]]-0.2,0},{Min[finsol[#]]+0.2,0}}],Arrow[{{Max[finsol[#]]+0.2,0},{6,0}}]}]}],PlotRange{{-5,6},{-7,5}},AspectRatio1]&/@Range[-5,5,0.5],3],ImageSize1200,Spacings{-10,0},FrameAll]
Out[]=
​
Now taking the arrows and fixed points alone and plotting them as a vector field:
We get the bifurcation diagram by rotating this whole plot above:
We can plot these fixed points and this gives us
We actually already have whether the fixed points are stable or unstable from the vector field plot above, so let’s put these on:
The critical points are when the term inside the square root vanishes (when we go from no fixed points, to one, to two):
We can actually rewrite the equation very simply to get it into Normal form. Let’s first of all complete the square:
Which is the normal form for a saddle-node bifurcation.
Homepage