The kicked rotor map, discussed in Lecture 35.  These commands can be used to produce all plots shown in class.

A non-chaotic regime.

nt=700;
JPhi=Table[0.0,{i,nt},{j,2}];
ϵ=0.05;
JPhi[[1,1]]=0.4;
JPhi[[1,2]]=0.4;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi1=JPhi;
JPhi=Table[0.0,{i,nt},{j,2}];
JPhi[[1,1]]=0.4;
JPhi[[1,2]]=2.2;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi2=JPhi;
JPhi=Table[0.0,{i,nt},{j,2}];
JPhi[[1,1]]=0.4;
JPhi[[1,2]]=5.1;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi3=JPhi;
ListPlot[{JPhi1,JPhi2,JPhi3},AxesLabel->{ϕ,J},PlotRange->All]

nt=500;
nstart=23;
JPhi=Table[0.0,{i,nt*nstart},{j,2}];
ϵ=0.05;
For[m=1,m<=nstart,m++,
tt=1+nt*(m-1);
JPhi[[tt,2]]=2*Pi/(nstart+0.3)*m;
JPhi[[tt,1]]=2*Pi/(nstart+0.3)*m;
For[t=2,t<= nt,t++,
tt=t+nt*(m-1);
JPhi[[tt,2]]=Mod[ϵ*Sin[JPhi[[tt-1,1]]]+JPhi[[tt-1,2]],2*Pi];
JPhi[[tt,1]]=Mod[JPhi[[tt,2]]+JPhi[[tt-1,1]],2*Pi]];
];
ListPlot[JPhi,AxesLabel->{ϕ,J},PlotRange->All]

Approaching transition to chaos.

nt=700;
JPhi=Table[0.0,{i,nt},{j,2}];
ϵ=0.6;
JPhi[[1,1]]=0.4;
JPhi[[1,2]]=0.4;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi1=JPhi;
JPhi=Table[0.0,{i,nt},{j,2}];
JPhi[[1,1]]=0.6;
JPhi[[1,2]]=2.2;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi2=JPhi;
JPhi=Table[0.0,{i,nt},{j,2}];
JPhi[[1,1]]=0.9;
JPhi[[1,2]]=3.81;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi3=JPhi;
ListPlot[{JPhi1,JPhi2,JPhi3},AxesLabel->{ϕ,J},PlotRange->All]

nt=500;
nstart=23;
JPhi=Table[0.0,{i,nt*nstart},{j,2}];
ϵ=0.6;
For[m=1,m<=nstart,m++,
tt=1+nt*(m-1);
JPhi[[tt,2]]=2*Pi/(nstart+0.3)*m;
JPhi[[tt,1]]=2*Pi/(nstart+0.3)*m*0;
For[t=2,t<= nt,t++,
tt=t+nt*(m-1);
JPhi[[tt,2]]=Mod[ϵ*Sin[JPhi[[tt-1,1]]]+JPhi[[tt-1,2]],2*Pi];
JPhi[[tt,1]]=Mod[JPhi[[tt,2]]+JPhi[[tt-1,1]],2*Pi]];
];
ListPlot[JPhi,AxesLabel->{ϕ,J},PlotRange->All]

Just past the onset of chaos.

nt=700;
JPhi=Table[0.0,{i,nt},{j,2}];
ϵ=1.05;
JPhi[[1,1]]=0.4;
JPhi[[1,2]]=0.4;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi1=JPhi;
JPhi=Table[0.0,{i,nt},{j,2}];
JPhi[[1,1]]=0.6;
JPhi[[1,2]]=2.2;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi2=JPhi;
JPhi=Table[0.0,{i,nt},{j,2}];
JPhi[[1,1]]=0.9;
JPhi[[1,2]]=3.81;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi3=JPhi;
ListPlot[{JPhi1,JPhi2,JPhi3},AxesLabel->{ϕ,J},PlotRange->All]

nt=500;
nstart=23;
JPhi=Table[0.0,{i,nt*nstart},{j,2}];
ϵ=1.05;
For[m=1,m<=nstart,m++,
tt=1+nt*(m-1);
JPhi[[tt,2]]=2*Pi/(nstart+0.3)*m;
JPhi[[tt,1]]=2*Pi/(nstart+0.3)*m*0;
For[t=2,t<= nt,t++,
tt=t+nt*(m-1);
JPhi[[tt,2]]=Mod[ϵ*Sin[JPhi[[tt-1,1]]]+JPhi[[tt-1,2]],2*Pi];
JPhi[[tt,1]]=Mod[JPhi[[tt,2]]+JPhi[[tt-1,1]],2*Pi]];
];
ListPlot[JPhi,AxesLabel->{ϕ,J},PlotRange->All]

Well into the chaotic regime.

nt=700;
JPhi=Table[0.0,{i,nt},{j,2}];
ϵ=1.8;
JPhi[[1,1]]=0.4;
JPhi[[1,2]]=0.4;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi1=JPhi;
JPhi=Table[0.0,{i,nt},{j,2}];
JPhi[[1,1]]=0.0;
JPhi[[1,2]]=3.0;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi2=JPhi;
JPhi=Table[0.0,{i,nt},{j,2}];
JPhi[[1,1]]=0.9;
JPhi[[1,2]]=3.81;
For[t=2,t<= nt,t++,
JPhi[[t,2]]=Mod[ϵ*Sin[JPhi[[t-1,1]]]+JPhi[[t-1,2]],2*Pi];
JPhi[[t,1]]=Mod[JPhi[[t,2]]+JPhi[[t-1,1]],2*Pi]];
JPhi3=JPhi;
ListPlot[{JPhi1,JPhi2,JPhi3},AxesLabel->{ϕ,J},PlotRange->All]

nt=500;
nstart=23;
JPhi=Table[0.0,{i,nt*nstart},{j,2}];
ϵ=1.8;
For[m=1,m<=nstart,m++,
tt=1+nt*(m-1);
JPhi[[tt,2]]=2*Pi/(nstart+0.3)*m;
JPhi[[tt,1]]=2*Pi/(nstart+0.3)*m*0;
For[t=2,t<= nt,t++,
tt=t+nt*(m-1);
JPhi[[tt,2]]=Mod[ϵ*Sin[JPhi[[tt-1,1]]]+JPhi[[tt-1,2]],2*Pi];
JPhi[[tt,1]]=Mod[JPhi[[tt,2]]+JPhi[[tt-1,1]],2*Pi]];
];
ListPlot[JPhi,AxesLabel->{ϕ,J},PlotRange->All]