Driven damped Pendulum

Poincare Sections for the Forced Damped Pendulum

System of differential equations:
driving frequency omega
driving parameter f
damping parameter beta

Clear[x,t,f,omega,beta]
omega=2/3;
f= 1.5;
beta =0.4;
equationx=x'[t]==xdot[t]
equationxdot=xdot'[t]==-2*beta* xdot[t]-Sin[x[t]] + f*Cos[omega*t]

Initial conditions, period, and number of periods for the trajectory

xstart = 1.;
xdotstart = 0.;
period = 2Pi/omega
Np=100

Attractor with period equal to that of the driving frequency

Plot the trajectory and the Poincare section

xrange = {-3.,3.};
xdotrange = {-3.,3.};

Solve the system of differential equations numerically for Np periods of the driving force,
saving the trajectory and the endpoint for each period

trajectorylist = {}
sectionlist = {{xstart,xdotstart}}
Do[
initialx=x[0]==xstart;
initialxdot=xdot[0]==xdotstart;
solution=NDSolve[{equationx,equationxdot,initialx,initialxdot},{xdot[t],x[t]},{t,0,period}];trajectory[t_]={x[t]/.solution[[1]],xdot[t]/.solution[[1]]};
trajectoryplot=ParametricPlot[trajectory[t],{t,0,period},PlotStyle->{{Red,Thickness[0.002]}},DisplayFunction->Identity];
trajectorylist=Append[trajectorylist,trajectoryplot];
{finalx,finalxdot} =trajectory[period];
sectionlist =Append[sectionlist,{finalx,finalxdot}];
{xstart,xdotstart}={finalx,finalxdot},
{n,1,Np}]

Plot the trajectory and the Poincare section

Show[trajectorylist,PlotRange->{xrange,xdotrange},AxesLabel->{"x","x'"},DisplayFunction->$DisplayFunction]
ListPlot[sectionlist,PlotRange->{xrange,xdotrange},AxesLabel->{"x","x'"},PlotStyle->{Blue,PointSize[0.01]},DisplayFunction->$DisplayFunction]

Change the plot ranges if necessary.  Then click on the last 2 cells again
to solve the system of differential equations for an additional Np periods
and to plot the phase space trajectory and the Poincare section of the attractor

First period doubling

Determine the value of beta at which the period of the attractor changes to 2 times the period of the driving force.

Draw the phase space trajectory and the Poincare section for an attractor whose period is 2 times that of the driving force.
Show them to the instructor.

Second period doubling

Determine the value of beta at which the period of the attractor changes to 4 times the period of the driving force.
Use Feigenbaum's constant to estimate the value of beta at which the period of the attractor changes to 8 times the period of the driving force.

Draw the phase space trajectory and the Poincare section for an attractor whose period is 4 times that of the driving force.
Show them to the instructor.

Third period doubling

Determine the value of beta at which the period of the attractor changes to 8 times the period of the driving force.
Use Feigenbaum's constant to estimate the value of beta at which the motion becomes chaotic.

Draw the phase space trajectory and the Poincare section for an attractor whose period is 8 times that of the driving force.
Show them to the instructor.