HenonMapEigenvalues.nb
To run a cell, put the cursor anywhere in the cell.  Then type Shift-Enter, or hit the Enter key on the bottom right of the keyboard.  
To see help for the commands, like Do, or anything that is capitalized, select the work and hit F1 (or go to the Help menu).

The a-b plane for the Henon Map
Saddle Node Curve is snCurve (red)
Period Doubling Curve is pdCurve (blue)
Hopf Bifurcation Curve is hopfCurve (green)

snCurve = ParametricPlot[{-1/4 (b-1)^2, b}, {b, -1.5,3}, PlotStyle -> Red];
pdCurve = ParametricPlot[{3/4 (b-1)^2, b}, {b, -1.3,3}, PlotStyle -> Blue];
hopfCurve = ParametricPlot[{a, -1}, {a, -1, 3}, PlotStyle -> Green];
abPlane =Show[hopfCurve, snCurve, pdCurve, Graphics[{Text["no fixed points", {-1,1}],Text["(SubPlus[v]) saddle\n(SubMinus[v]) saddle", {2,1}],Text["(SubPlus[v]) saddle\n(SubMinus[v]) source", {.7,2.7}],Text["(SubPlus[v]) sink\n(SubMinus[v]) saddle", {.5,-.5}],Text["(SubPlus[v]) source\n(SubMinus[v]) saddle", {.7,-1.4}]}],PlotRange -> All, AxesLabel ->{"a","b"}]

The next cell includes “test points”

a = .7; b =2.7; (* upper region *)
a = 2; b =1.; (* right region *)
a = -1; b =0.8; (* left region *)
a = 0.7; b =-1.4; (* bottom lower region *)
a = 0.; b =1.; (* red-blue intersection *)
a = -1.; b =-1.; (* red-green intersection *)
a = 3.; b =-1.; (* blue-green intersection *)
a = 0.5; b =-0.5; (* middle lower region *)

xp = ((b-1) + Sqrt[(1-b)^2 + 4a])/2;
xm = ((b-1) - Sqrt[(1-b)^2 + 4a])/2;
Dfp= {{-2xp, b}, {1,0}};
Dfm= {{-2xm, b}, {1,0}};
Print["xp = ", xp, ", xm = ", xm];
Print["Eigenvalues of fixed point (SubPlus[v]) =(xp, xp) are ", Eigenvalues[Dfp]];
Print["Eigenvalues of fixed point (SubMinus[v]) =(xm, xm) are ", Eigenvalues[Dfm]];
Show[abPlane, Graphics@Point[{a, b}], PlotRange -> All]

runHenon := Module[{},
xp = ((b-1) + Sqrt[(1-b)^2 + 4a])/2;
xm = ((b-1) - Sqrt[(1-b)^2 + 4a])/2;
Dfp= {{-2xp, b}, {1,0}};
Dfm= {{-2xm, b}, {1,0}};
title = "Henon map, (x, y) ↦ (a - (x^2) + b y, x)\n a = "<>ToString[a]<>", b = "<>ToString[b];
fofxy[x_, y_] := {a - x^2 + b y,  x}; (* the Henon map. *);

figLeft = Show[abPlane, Graphics@Point[{a, b}], PlotRange -> All];
Print["xp = ", xp, ", xm = ", xm];
Print["Eigenvalues of fixed point (SubPlus[v]) =(xp, xp) are ", Eigenvalues[Dfp]];
Print["Eigenvalues of fixed point (SubMinus[v]) =(xm, xm) are ", Eigenvalues[Dfm]];
ic = {xp +.01, xp- .01};
f[v_] := fofxy[v[[1]], v[[2]] ];  (* maps points (lists) to points *)
pt = ic;
Do[pt = N@f[pt]
, {nTransients}
];
orbit = {pt};
Do[pt = f[pt];
AppendTo[orbit, pt]
, {nIterates}
];
figRight = ListPlot[orbit, AspectRatio -> Automatic, Frame -> True, Axes -> False, PlotStyle -> PointSize[pointSize], PlotLabel ->title, Prolog ->{Red, PointSize[Large], Point[{xp, xp}]}];
GraphicsRow[{figLeft, figRight}, ImageSize -> 800]
]

The next cell runs the Henon Map.  A red dot is put at the fixed point (SubPlus[v]) = (xp, xp).
One sequence worth trying is with b = -0.3:
a = 1.2, 1.4, 1.9, 1.93, 1.95, 1.951, 1.952, 1.953, 1.954, 1.955
Standard Parameters are a = 1.4, b = 0.3

a = 1.4;
b =.3;
nTransients = 1000;
nIterates = 10000;  (*Don't make this much larger than 50 000. It gets slow. *)
pointSize = Tiny; (* Large, Medium, Small, or Tiny *)

runHenon

The next cell can be used to show self-similarity near the “positive” fixed point (SubPlus[v]) as in Henon’s original paper.  You to make nIterates large in the previous cell to see much when you zoom in.

nZoom = 0;  (*0, 1, or 2 are best *)

eval = 0.17647; w0 = 0.2;  (* Absolute value of stable eigenvalue for v+ at a = 2.1, b = -0.3 *)
eval = 0.155946; w0 = 0.35;  (* Absolute value of stable eigenvalue for v+ at a = 1.4, b = 0.3 *)
width = w0 eval^nZoom;
Show[figRight, PlotRange ->{{xp-width, xp+width}, {xp - width, xp + width}}]