The dissipative Henon map

Definition

Clear["Global`*"]
b=0.3;
a=0.8;  (*0.8, 1.05, 1.2, 1.3, 1.4*)
g1[x_,y_]=1+y-a x^2
g2[x_,y_]=b x
Print["x1=",g1[x0,y0],"   y1=",g2[x0,y0]]
(* Jacobian *)
M[x_,y_]={{D[g1[x,y],x], D[g1[x,y],y]},{D[g2[x,y],x],D[g2[x,y],y]}};
MatrixForm[M[x,y]]
Det[M[x,y]]

Fixed points

fp=Solve[{x==g1[x,y],y==g2[x,y]},{x,y}]

Stability

(* fixed point 1 *)
x01=x/.fp[[1]]; y01=y/.fp[[1]];
M1=M[x01,y01];
L1=Eigenvalues[M1]
(* fixed point 2 *) 
x02=x/.fp[[2]]; y02=y/.fp[[2]]; 
M2=M[x02,y02];
L2=Eigenvalues[M2]
(* plot of fixed points *)
pt1=ListPlot[{{x01,y01}},PlotStyle->{PointSize[0.02],Red}];
pt2=ListPlot[{{x02,y02}},PlotStyle->{PointSize[0.02],Red}];
Show[pt1,pt2]

Run Orbit

x0=x01; y0=y01+0.0001;
trajectory={{x0,y0}};
xEvolution={x0};
n=6000;
For[i=1,i<n,i++, x1=g1[x0,y0]; y1=g2[x0,y0]; AppendTo[xEvolution,x1];AppendTo[trajectory,{x1,y1}]; x0=x1;y0=y1;]
plotevol=ListPlot[xEvolution]
plottraj=ListPlot[trajectory];
Short[trajectory,6]
Show[pt1,pt2, plottraj,PlotRange->All, Frame->True]

limittraj=Drop[trajectory, n/2];
plottraj=ListPlot[limittraj];
Show[pt1,pt2, plottraj,PlotRange->All, Frame->True]
Short[limittraj,4]

Period - 2

g21[x_,y_]=g1[g1[x,y],g2[x,y]] //Simplify
g22[x_,y_]=g2[g1[x,y],g2[x,y]] //Simplify
p2=Solve[{x==g21[x,y],y==g22[x,y]},{x,y}]

Stability

k=2; (* orbit *)
x0=x/.p2[[k]]
y0=y/.p2[[k]]
MJ2[x_,y_]={{D[g21[x,y],x], D[g21[x,y],y]},{D[g22[x,y],x],D[g22[x,y],y]}};
L1=Eigenvalues[MJ2[x0,y0]]