Application of the Poincaré-Bendixson theorem

Ex. 1

Clear[r,μ]
tmax=3;

Manipulate[
polarvectorfield={r (1-r^2)+μ r Cos[θ],1};
rmax=Sqrt[1+μ];
rmin=Sqrt[1-μ];
plotrmax=PolarPlot[rmax,{θ,0,2 π},PlotStyle->{Thick,Black}];
plotrmin=PolarPlot[rmin,{θ,0,2 π},PlotStyle->{Thick,Black}];
cartesianfield=TransformedField["Polar"->"Cartesian",polarvectorfield,{r,θ}->{x,y}];
sp0=StreamPlot[cartesianfield,{x,-1.7,1.7},{y,-1.7,1.7},FrameLabel->{x[t],y[t]},PlotLabel->Style["Full equation",Large],ImageSize->550,LabelStyle->20,StreamColorFunction->None,StreamStyle->Gray];
sp=StreamPlot[cartesianfield,{x,-1.7,1.7},{y,-1.7,1.7},FrameLabel->{x[t],y[t]},PlotLabel->Style["Full equation",Large],StreamPoints->{{-1,0}},ImageSize->550,LabelStyle->20,StreamColorFunction->None,StreamStyle->{Thick,Blue}];
Show[sp0, sp,plotrmax,plotrmin],
{{μ,0.5,"Parameter μ"},0,1.5,Appearance->"Labeled"},SaveDefinitions->True,TrackedSymbols:>{μ}]

Phase plots, selected

Clear[r]
tmax=3;
μ=0.16;(*smaller then 1*)

polarvectorfield0={r (1-r^2),1};
polarvectorfield={r (1-r^2)+μ r Cos[θ],1};
rmax=Sqrt[1+μ];
rmin=Sqrt[1-μ];

plotrmax=PolarPlot[rmax,{θ,0,2 π},PlotStyle->{Thick,Black}];
plotrmin=PolarPlot[rmin,{θ,0,2 π},PlotStyle->{Thick,Black}];

fixedp=Graphics[{EdgeForm[{Thick,Red}],FaceForm[White],Disk[{0,0},{0.035,0.035}]}];

cartesianfield0=TransformedField["Polar"->"Cartesian",polarvectorfield0,{r,θ}->{x,y}];
sp00=StreamPlot[cartesianfield0,{x,-1.3,1.3},{y,-1.3,1.3},FrameLabel->{x[t],y[t]},PlotLabel->Style["Case where μ = 0",Large],ImageSize->510,LabelStyle->20,StreamColorFunction->None,StreamStyle->Gray];
sp0=StreamPlot[cartesianfield0,{x,-1.3,1.3},{y,-1.3,1.3},FrameLabel->{x[t],y[t]},PlotLabel->Style["Case where μ = 0",Large],StreamPoints->{{1,0}},ImageSize->510,LabelStyle->20,StreamColorFunction->None,StreamStyle->{Thick,Blue}];
pp0=Show[sp00, sp0,plotrmax,plotrmin,fixedp];

cartesianfield=TransformedField["Polar"->"Cartesian",polarvectorfield,{r,θ}->{x,y}];
sp000=StreamPlot[cartesianfield,{x,-1.3,1.3},{y,-1.3,1.3},FrameLabel->{x[t],y[t]},PlotLabel->Style["Full equation μ < 1",Large],ImageSize->510,LabelStyle->20,StreamColorFunction->None,StreamStyle->Gray];
sp=StreamPlot[cartesianfield,{x,-1.3,1.3},{y,-1.3,1.3},FrameLabel->{x[t],y[t]},PlotLabel->Style["Full equation μ < 1",Large],StreamPoints->{{-1,0}},ImageSize->510,LabelStyle->20,StreamColorFunction->None,StreamStyle->{Thick,Blue}];
pplt=Show[sp000, sp,plotrmax,plotrmin,fixedp];
Row[{pplt,pp0}]

Decoupled 1-D phase portrait (μ = 0)

Clear[r]
Plot[r (1-r^2),{r,0,1.2},AxesLabel->{r,"(∂_t)r"},ImageSize->500,LabelStyle->20]

Creation of this notebook was supported by IT Academy program of Information Technology Foundation for Education (HITSA).