WOLFRAM NOTEBOOK

MAM2046W - Second year nonlinear dynamics

Section 3.4: Global bifurcations of cycles

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We’ve now seen that while the normal types of bifurcations from 1d persist in 2d and don’t give us anything very different, the existence of limit cycles gives us something more interesting. So far we’ve looked at Supercritical Hopf bifurcations, by which a stable fixed point can explode out into a stable limit cycle and an unstable fixed point, or in the case of the subcritical Hopf bifurcation, an unstable limit cycle can engulf a stable fixed point to become an unstable fixed point. Finally we saw degenerate Hopf bifurcations where there are no limit cycles but only centers about the fixed point which changes from a stable to an unstable spiral across the bifurcation.
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Just as a reminder, let’s see these three in action.
In[]:=
eqns=SimplifySolveD
2
x[t]
+
2
y[t]
,t==μ
(
2
x[t]
+
2
y[t]
)
+
3
(
2
x[t]
+
2
y[t]
)
-
5
(
2
x[t]
+
2
y[t]
)
,DArcTan
y[t]
x[t]
,t==0.1+0.1(
2
x[t]
+
2
y[t]
),{x'[t],y'[t]}[[1]]//FullSimplify;​​vf[μ_]=({x'[t],y'[t]}/.eqns)/.x[t]->x/.y[t]->y;​​pl[μ_]:=ShowVectorPlot[vf[μ],{x,-1.5,1.5},{y,-1.5,1.5}],IfIm
1
2
1-
1+4μ
==0,GraphicsThick,Circle{0,0},
1
2
1+
1+4μ
,GraphicsDashed,Thick,Circle{0,0},
1
2
1-
1+4μ
,Graphics[Circle[{0,0},0]],If[μ<0,Graphics[Disk[{0,0},0.05]],Graphics[Circle[{0,0},0.05]]],PlotLabel->Style["μ = "<>ToString[μ],20],ImageSize->500​​Labeled[GraphicsGrid[Partition[pl[#]&/@{-0.2,-0.1,-0.05,0.1},2],ImageSize->700],Style["Subcritical Hopf bifurcation happens at μ=0",20]]​​
Out[]=
Subcritical Hopf bifurcation happens at μ=0
In[]:=
eqns=SimplifySolveD
2
x[t]
+
2
y[t]
,t==μ
(
2
x[t]
+
2
y[t]
)
-
3
(
2
x[t]
+
2
y[t]
)
,DArcTan
y[t]
x[t]
,t==0.1+0.1(
2
x[t]
+
2
y[t]
),{x'[t],y'[t]}[[1]]//FullSimplify;​​vf[μ_]=({x'[t],y'[t]}/.eqns)/.x[t]->x/.y[t]->y;​​pl[μ_]:=Show[VectorPlot[vf[μ],{x,-1.5,1.5},{y,-1.5,1.5}],If[Im[
μ
]==0,Graphics[Dashed,Thick,Circle[{0,0},
μ
]],Graphics[Thick,Circle[{0,0},
μ
]],Graphics[Circle[{0,0},0]]],If[μ<0,Graphics[Disk[{0,0},0.05]],Graphics[Circle[{0,0},0.05]]],PlotLabel->Style["μ = "<>ToString[μ],20],ImageSize->500]​​Labeled[GraphicsGrid[Partition[pl[#]&/@{-0.2,0,0.25,1},2],ImageSize->700],Style["Supercritical Hopf Bifurcation occurs at μ=0",20]]​​
Out[]=
Supercritical Hopf Bifurcation occurs at μ=0
In[]:=
Labeled[GraphicsGrid[Partition[Table[StreamPlot[{y,-μy-Sin[x]},{x,-3,3},{y,-3,3},ImageSize->500,PlotLabel->Style["μ = "<>ToString[μ],20]],{μ,-1,2,1}],2],ImageSize->700],Style["Degenerate Hopf bifurcation happens at μ=0.Centers appear at μ=0, before and after we have stable and unstable spirals",20]]​​
Out[]=
Degenerate Hopf bifurcation happens at μ=0.Centers appear at μ=0, before and after we have stable and unstable spirals
So, what else can happen beyond these bifurcations? For the first case we think about what can happen with a flow on the line, and in particular a saddlepoint bifurcation. This occurs when a stable and unstable fixed point appear out of the blue and move away from one another (generically). An example of this would be
Which has phase portraits:
Where we are interested in the red box in the bifurcation diagram:
What has this got to do with limit cycles? Well, how about the vector field:
So this system exhibits both a subcritical and saddlenode Hopf bifurcation, we can see them both happening here
How about the stability of these? Well, we can plot the second equation as a simple flow on the one-dimensional line and see:
Now plotting this along with the nullclines in polar coordinates we have:
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Can we calculate the period of the oscillation by integrating the differential equation:
The period is how long θ takes to get from 0 to 2π, so we write:
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Deep breath...we have one more limit cycle bifurcation to go, and that is a Homoclinic (or saddle-loop) Bifurcation. The ingredients needed for this are a saddle, and a limit cycle. When the limit-cycle hits the saddle, the bifurcation occurs. It’s not easy to find simple examples of this, so the following simply illustrates what happens in the case of
precisely at the bifurcation point (the middle point), we get a homoclinic orbit.
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In the next section we’re going to talk about coupled oscillators and periodicity, and then we will start edging closer to chaos...
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