Henon Map

henon[a_, b_][{x_, y_}]:={a - x^2 + b*y, x}
a1= .9;
b1 = .4;
x0=.3;
y0=.3;
numiterates = 800;
beginplotting = 1;

ListofPoints[a_, b_]:=Table[Nest[henon[a, b], {x0, y0}, k], {k, beginplotting, numiterates}]
ListPlot[ListofPoints[a1, b1], PlotStyle->PointSize[Large]]

Here are the time series for the Henon map for a variety of parameters {(.1, .4), (.2, .4), (.3, .4), (.4, .4), (.5, .4)}.

ListofPoints[a_, b_]:=Table[Nest[henon[a, b], {x0, y0}, k], {k, beginplotting, numiterates}]
ListPlot[{ListofPoints[.1, .4],ListofPoints[.2, .4], ListofPoints[.3, .4], ListofPoints[.4, .4], ListofPoints[.5, .4]}, PlotStyle->PointSize[Large]]

For the fixed parameter b = .4, we can vary the parameter “a” and plot the list of iterates following the initial transient behavior.

Manipulate[ListPlot[Table[Style[Nest[henon[a, .4], {x0, y0}, k], Darker[Hue[k/50]]], {k, 200, 250}], PlotLabel->Style[Framed["= a " a], Bold, Black]], {a, 0, 1.25, .0001}]Manipulate[ListPlot[Table[Style[Nest[henon[a, .4], {x0, y0}, k], Darker[Hue[k/50]]], {k, 200, 600}], PlotLabel->Style[Framed["= a " a], Bold, Black]], {a, .8, 1.248, .0001}]

Some pairs of parameters will give overflow errors because the orbits are escaping to infinity.

Manipulate[ListPlot[Table[Style[Nest[henon[a, b], {x0, y0}, k], Darker[Hue[k/50]]], {k, 200, 250}], PlotLabel->Style[Framed[{"= a " a, "=b " b}], Bold, Black]], {a, 0, 1.8, .0001}, {b, .1, .5}]