Math332. Partial notes for Week 4

Periodic orbits and their stability

<<H:\.WIN2k_SETTINGS\DESKTOP\GraphicalAnalysis-v10.m

SetDirectory["~/Documents/FoS-Courses/MATH332/Notebooks/"]

(* Unprotect[Iterate,PlotIterates,BifIterate,Bifurcation];  *)
<<GraphicalAnalysis-v10.m

Period 2 orbits of the logistic equation

g[x_] := 2.9x(1-x) ;Plot[{x,g[x],g[g[x]]},{x,0,1},PlotStyle->({Hue[#/3]}&/@Range[0,2]),Background->White]

g[x_] := 3.2x(1-x) ;Plot[{x,g[x],g[g[x]]},{x,0,1},PlotStyle->({Hue[#/3]}&/@Range[0,2]),Background->White]

f[x_,a_] := a x (1-x)

Bifurcation[f[x,a],{x,0.1, 0.5, 0.9},{a,2.9,3.2}, Iterations -> 2000, Transients->1900, Slices-> 300, Frame->True,GridLines->{{3},{0}},Background->White]

Period 3 orbits of the logistic equation

g[x_] := 3.833x(1-x) ;Plot[{x,g[x],g[g[g[x]]]},{x,0,1}
,Frame->True,PlotStyle->({Hue[#/3]}&/@Range[0,2]),Background->White]

PlotIterates[g[x],{x,0,1},{0.01, 0.1, 0.3, 0.5, 0.7,0.9}, Iterations->1000,Background->White,Frame->True]

f[x_,a_] := a x (1-x)

Show[Bifurcation[f[x,a],{x,0.1,0.2,0.3,0.4,0.5,0.7, 0.8,0.9},{a,3.83,3.845}, Iterations -> 600, Transients->500, Slices-> 300], PlotRange->{0.4,0.6},Background->White]

g[x_] := 3.84x(1-x) ;

Plot[{x,g[x],g[g[x]]},{x,0,1}
,Frame->True,PlotStyle->({Hue[#/3]}&/@Range[0,2]),Background->White]

Plot[{x,g[g[g[g[x]]]],g[g[g[g[g[x]]]]]},{x,0,1}
,Frame->True,PlotStyle->({Hue[#/3]}&/@Range[0,2]),Background->White]