Lyapunov Exponents And Cobweb Diagrams for Logistic Map

For the logistic map, we can compute the Lyapunov exponent for different values of r at the initial condition x0=1/2.  On the right is the plot of the Lyapunov Exponent with the value r along the x-axis.  Remember that if LE(f) < 0, then the system is not chaotic, and if LE(f) > 0, then the system is chaotic on the interval containing the given orbit.  

Calculating the Lyapunov Exponent is computationally taxing, so I pre-loaded the value of LE(f_r) for r from 0 to 1 and r from 3.5 to 4 at the bottom of this worksheet.

f[r_][x_]:=r*x*(1-x)

LNum[r_]:=GeometricMean[Table[Abs[D[f[r][x], x]/.x-> Nest[f[r], .5, k]], {k, 1, 950}]]/.Indeterminate->-3.0

LExp[r_]:=Log[GeometricMean[Table[Abs[D[f[r][x], x]/.x-> Nest[f[r], .5, k]], {k, 1, 950}]]/.Indeterminate->-3.0]/.Indeterminate->-3.0

LExp[2.5]

LExp[3.9998]

ListofIterates[r_]:=Flatten[Table[{{Nest[f[r], .5, k], Nest[f[r], .5, k]}, {Nest[f[r], .5, k], Nest[f[r], .5,  k+1]}}, {k, 10, 100}], 1]
PlotK[k_]:=ListPlot[{TableofLyap, {{k, LExp[k]}}}, PlotStyle->{{Darker[Blue], PointSize[.007]}, {Red, PointSize[.02]}}]
PlotKThreeFive[k_]:=ListPlot[{TableofLyapThreeFive, {{k, LExp[k]}}}, PlotStyle->{{Darker[Blue], PointSize[.007]}, {Red, PointSize[.02]}}]

Manipulate[GraphicsRow[{Plot[{f[r][x], x}, {x,0, 1},PlotStyle->{{Thick, Red}, {Dashed, Thick,  Black}},  Prolog->{Darker[Blue], Thickness[.004], Line[ListofIterates[r]]}], PlotK[r]}], {r, 1, 3.99999, .001}]

Manipulate[GraphicsRow[{Plot[{f[r][x], x}, {x,0, 1},PlotStyle->{{Thick, Red}, {Dashed, Thick,  Black}},  Prolog->{Darker[Blue], Thickness[.004], Line[ListofIterates[r]]}], PlotKThreeFive[r]}], {r, 3.5, 3.99999, .0001}]

TableofLyap={{1.`,-0.011910336369651565`},{1.01`,-0.017273351544519457`},{1.02`,-0.026024756619489615`},{1.03`,-0.035480433071044364`},{1.04`,-0.045286894490393`},{1.05`,-0.055335168332500836`},{1.06`,-0.06557814634512756`},{1.07`,-0.07599191020453082`},{1.08`,-0.08656325365622122`},{1.09`,-0.09728463200679288`},{1.1`,-0.10815179608128732`},{1.11`,-0.11916256334806058`},{1.12`,-0.13031612872317097`},{1.13`,-0.14161265451257343`},{1.1400000000000001`,-0.153053014864314`},{1.15`,-0.1646386305934524`},{1.16`,-0.17637135935357853`},{1.17`,-0.18825342106192428`},{1.18`,-0.20028734656429537`},{1.19`,-0.21247594209910356`},{1.2`,-0.22482226481098988`},{1.21`,-0.2373296062029748`},{1.22`,-0.25000148144398454`},{1.23`,-0.26284162311101433`},{1.24`,-0.27585397838267484`},{1.25`,-0.2890427089963563`},{1.26`,-0.3024121934851899`},{1.27`,-0.3159670313547`},{1.28`,-0.3297120489624683`},{1.29`,-0.3436523069402537`},{1.3`,-0.3577931090553503`},{1.31`,-0.37214001245238076`},{1.32`,-0.38669883925213544`},{1.33`,-0.40147568951331514`},{1.34`,-0.4164769555880908`},{1.35`,-0.4317093379247368`},{1.3599999999999999`,-0.44717986239135343`},{1.37`,-0.4628958992147059`},{1.38`,-0.47886518364819997`},{1.3900000000000001`,-0.4950958385036461`},{1.4`,-0.5115963987031658`},{1.4100000000000001`,-0.5283758380310792`},{1.42`,-0.5454435982911987`},{1.43`,-0.5628096211034394`},{1.44`,-0.5804843826054054`},{1.45`,-0.5984789313605264`},{1.46`,-0.6168049298150167`},{1.47`,-0.635474699692392`},{1.48`,-0.6545012717675366`},{1.49`,-0.673898440523599`},{1.5`,-0.6936808242658044`},{1.51`,-0.7138639313483444`},{1.52`,-0.7344642332660049`},{1.53`,-0.75549924547364207504482295194133022635`18.81054741041989},{1.54`,-0.77698761692715047764300621248882218891`18.822727472861125},{1.55`,-0.79894922949297874064986734996368960125`18.834832557733005},{1.56`,-0.8214053085541805750334592662714636583`18.846870880752217},{1.57`,-0.84437854635512652073886402995926904122`18.858850565868728},{1.58`,-0.86789323988142738856198452630527475359`18.870779681091847},{1.5899999999999999`,-0.89197544537505141307998120093064736835`18.882666274603846},{1.6`,-0.91665315194799374766439578911317556452`18.89451841159662},{1.6099999999999999`,-0.94195647719475910922451404974311913338`18.906344212295213},{1.62`,-0.96791788823174086392707259226849172913`18.918151891672107},{1.63`,-0.9945724522318609753209406225232128269`18.929949801410267},{1.6400000000000001`,-1.02195812130368473617455832082297111789`18.94174647474296},{1.65`,-1.05011605752093228676199934826436905602`18.953550674888433},{1.6600000000000001`,-1.0790910050871457892568235949212245827`18.965371447911178},{1.67`,-1.10893171808059000120525646565198709968`18.977218180985393},{1.6800000000000002`,-1.13969145404465963617306182456349800792`18.989100667217286},{1.69`,-1.17142854597161163089895702635140887906`19.001029178411418},{1.7000000000000002`,-1.2042070681093323737783074056131757691`19.013014547455672},{1.71`,-1.2380976146846438322225870600503301882`19.025068262367217},{1.72`,-1.27317821532988158090465815856368391314`19.037202574512257},{1.73`,-1.30953541705907595844180065192655124783`19.049430624117637},{1.74`,-1.34726557053071174659884974971458624131`19.06176658697708},{1.75`,-1.38647636870271604360225672937937140507`19.07422584728088},{1.76`,-1.42728869974326934536592329701975944838`19.086825202851056},{1.77`,-1.46983889450801683546884020406024185185`19.099583110867144},{1.78`,-1.51428147390858643139907076280592715148`18.81148998893197},{1.79`,-1.56079253583203686678849752910198053889`18.82462855928363},{1.8`,-1.60957396902480059422891723861493894729`18.837994319583164},{1.81`,-1.66085874873238299215876825187253672838`18.851616078339145},{1.82`,-1.71491766545336632679961079093268595965`18.865526653875342},{1.83`,-1.77206797896769414069271183408242987998`18.879763757820783},{1.8399999999999999`,-1.83268469902204809886464224980522489425`18.894371133783313},{1.85`,-1.89721551153648437001232842153811680729`18.909400046579577},{1.8599999999999999`,-1.96620086233955343597404530653200905942`18.92491126205551},{1.87`,-2.04030149676400902013183909167222927731`18.940977727980012},{1.88`,-2.12033704488404299304086158356793668545`18.95768828088556},{1.8900000000000001`,-2.20734143591429721901464558354510145148`18.975152895698187},{1.9`,-2.30264479896091479164580924463060241588`18.993510329613066},{1.9100000000000001`,-2.40799866285198394876320400098704506292`19.01293962124046},{1.92`,-2.52577520769664788736458350190980987196`19.033678075836796},{1.9300000000000002`,-2.65930026659696607608159165985466086309`19.056050756923216},{1.94`,-2.8134447660824590811511998561855871163`19.080521773263207},{1.9500000000000002`,-2.9957602923690132001643233249177775836`18.806760444225294},{1.96`,-3.21889795961401933061310752540766602322`18.83796059374245},{1.97`,-3.50657429118601642540349684094791926405`18.8751364285589},{1.98`,-3.91203379849812912550600797837147663928`18.922655982424363},{1.99`,-4.60517551537949299344537222652702676343`18.993499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