In[938]:= 
lambda =0.3
k = 10; 
b =1.
Clear[m]
f1[x_,0] = x
f1[x_,m_] = b + (x-1)(1 - lambda) - ArcTan[m (x-1)]+1
f1[x_,m_,n_]:= f1[f1[x,m],m,n-1];f1[x_,m_,1] = b + x(1 - lambda) - ArcTan[m x]
Table[Plot[{x,f1[x,8,n]},{x,0,2}],{n,1,20}]
Plot[Evaluate[Table[f1[1,m,n],{n,15,20}]],{m,1,15}]
Table[Plot[{x,f1[x,11.18,n]},{x,1.25,1.35}],{n,1,3}]
NSolve[f1[x,11.19,3]==x,x,Reals]
In[1824]:= Table[Plot[f1[1,m,n],{m,1,10}],{n,1,20}]
Plot[Evaluate[Table[f1[1,m,n],{n,15,20}]],{m,1,10}]
In[1035]:= 

k = 1000
m = Range[20.,45,25.0/(k-1)];
a=1.
lambda =0.3
rhs[x_?VectorQ]:= a + (x-1)(1 - lambda) - ArcTan[m (x-1)]+1;
iterates = RecurrenceTable[{x[n+1]==rhs[x[n]],x[0]==ConstantArray[1.3,k]},x,{n, 10^4, 2 10^4 }];
data = Transpose[Ceiling[iterates 300 ]];
Transpose[{data[[1]], ConstantArray[1,Length[ data[[1]]  ]]} ];
count[data_, i_] := Module[{c, j},
{j, c} = Transpose[Tally[data]];
Transpose[{j, ConstantArray[i,Length[j]  ]}]->Log[N[c]]];
S = SparseArray[Table[count[data[[i]], i], {i, k}], k];
ArrayPlot[Reverse[S], FrameTicks->All]
In[260]:= ListPlot[data[[401]]]


k = 1000
a = Range[-2.,2.,4./(k-1)];
m=7.
lambda =0.3
rhs[x_?VectorQ]:= a + (x-4)(1 - lambda) - ArcTan[m (x-4)]+4;
iterates = RecurrenceTable[{x[n+1]==rhs[x[n]],x[0]==ConstantArray[3.,k]},x,{n, 10^4, 2 10^4 }];
data = Transpose[Ceiling[iterates 100 ]];
Transpose[{data[[1]], ConstantArray[1,Length[ data[[1]]  ]]} ];
count[data_, i_] := Module[{c, j},
{j, c} = Transpose[Tally[data]];
Transpose[{j, ConstantArray[i,Length[j]  ]}]->Log[N[c]]];
S = SparseArray[Table[count[data[[i]], i], {i, k}], k];
ArrayPlot[Reverse[S], FrameTicks->All]