pt = Plot[Log@Abs[a],{a, 0, 1}, PlotStyle -> Red];

ptDashed = Plot[Log@Abs[a],{a, 1, 4}, PlotStyle -> {{Red, Dotted}}];

p1 = Plot[Log@Abs[a-2], {a, 1, 3}, Epilog->{PointSize[Large],Point[{4, Log[2]}]}, PlotStyle -> Blue];

p1Dashed = Plot[Log@Abs[a-2], {a, 3, 4}, PlotStyle -> {{Blue, Dotted}}];

p2= Plot[1/2Log@Abs[4 +2a - a ^2], {a, 3, 1 + Sqrt[6]}, PlotStyle -> Brown];

p2Dashed= Plot[1/2Log@Abs[4 +2a - a ^2], {a,  1 + Sqrt[6], 4}, PlotStyle -> {Brown, Dotted}];

figLyap = Show[{pt,ptDashed,p1, p1Dashed, p2, p2Dashed,Graphics[{PointSize[Large], Point[{4, Log[2]}]}]
},  PlotRange -> {-2.5, 1}];

mt = Plot[Abs[a],{a, 0, 1}, PlotStyle -> Red];

mtDashed = Plot[Abs[a],{a, 1, 4}, PlotStyle -> {{Red, Dotted}}];

m1 = Plot[Abs[a-2], {a, 1, 3}, Epilog->{PointSize[Large],Point[{4, Log[2]}]}, PlotStyle -> Blue];

m1Dashed = Plot[Abs[a-2], {a, 3, 4}, PlotStyle -> {{Blue, Dotted}}];

m2= Plot[Sqrt@Abs[4 +2a - a ^2], {a, 3, 1 + Sqrt[6]}, PlotStyle -> Brown];

m2Dashed= Plot[Sqrt@Abs[4 +2a - a ^2], {a,  1 + Sqrt[6], 4}, PlotStyle -> {Brown, Dotted}];

figLyapMult = Show[{mt,mtDashed,m1, m1Dashed,m2, m2Dashed,Graphics[{PointSize[Large], Point[{4,2}]}]
},  PlotRange -> {-0, 2.0}];

bt = Plot[0, {a, 0,1}, PlotStyle -> {Thick,Red},  PlotRange ->{-0.01, 1}];

btDahsed = Plot[0, {a, 1, 4}, PlotStyle -> {Thick,Red,Dotted},  PlotRange ->{-0.01, 1}];

b1 = Plot[(a-1)/a, {a, 1, 3}, PlotStyle -> {Blue},  PlotRange ->{-0.01, 1}];

b1Dashed= Plot[(a-1)/a, {a, 3, 4}, PlotStyle -> {Blue, Dotted},  PlotRange ->{-0.01, 1}];

b2 = Plot[{(1 + a + Sqrt[a^2 -2a-3])/(2a), (1 + a - Sqrt[a^2 -2a-3])/(2a)}, {a, 3, 1+Sqrt[6]}, PlotStyle -> { Brown, Brown},  PlotRange ->{-0.01, 1}];

b2Dashed = Plot[{(1 + a + Sqrt[a^2 -2a-3])/(2a), (1 + a - Sqrt[a^2 -2a-3])/(2a)}, {a,1+Sqrt[6], 4}, PlotStyle -> {{ Brown, Dotted}},  PlotRange ->{-0.10, 1}];

figBifDiag = Show[bt, btDahsed, b1, b1Dashed, b2, b2Dashed, PlotRange ->{{0, 4}, {-0.01, 1}}];

figh = Magnify[GraphicsColumn[{figLyap, figBifDiag}], 1]

figL = Magnify[GraphicsColumn[{figLyapMult, figBifDiag}], 1]

figLh = Magnify[GraphicsGrid[{{figLyap, figLyapMult},{figBifDiag, figBifDiag}}, ImageSize -> 600], 1]

The upper left figure shows the Lyapunov exponent as a function of a, and the upper right figure shows the Lyapunov multiplier as a function of a.  The bottom diagrams both show a partial bifurcation diagram, the period 1 and 2 points as a function of a.

Export["/Users/jimswift/Dropbox/classes/MAT 667 2017.1/LyapunovExponentPlots/knownLiapExp.gif", figh, ImageSize -> 450]

Export["/Users/jimswift/Dropbox/classes/MAT 667 2017.1/LyapunovExponentPlots/knownLiapNum.gif", figL, ImageSize -> 450]

Export["/Users/jimswift/Dropbox/classes/MAT 667 2017.1/LyapunovExponentPlots/knownLiapExpNum.gif", figLh, ImageSize -> 900]