Visualizing Chaos in the
Physical Pendulum

In the laboratory we will investigate more of the behavior of the damped, non-linear, physical pendulum that was developed in the previous laboratory.

1. The Model.

The equation describing the physical pendulum can be written in terms of torques as

                                (Subscript[τ, total]) = Iα =  \(TraditionalForm\`τ\_gravity\  + \ τ\_friction\  + \ τ\_drive\)
                                                
where I is the moment of inertia, and α is the angular acceleration. The components of the total torque are summarized below.

                                              (Subscript[τ, gravity]) =    (-mg sinθ ) (Subscript[L, cm]) 
                                              
                                       (Subscript[τ, friction]) = -(Dd(θ/dt)) (Subscript[L, cm])
                                       
                                       (Subscript[τ, drive]) = (Subscript[L, cm])(Subscript[F, D])sin(Ωt)

In the first expression in this list (Subscript[L, cm]) is the distance from the pivot to the center of mass, m is the mass, and  θ is the angle with respect to the vertical. In the second formula D is the drag coefficient detemined by the nature of the physical pendulum. The minus sign here is critical since this torque must oppose the motion. For the driving term, the last in the list,   (Subscript[F, D]) is the amplitude of the force and Ω is the angular frequency of the driving force.

The pendulum is a uniform cylinder that is pivoted at one end. The following inital conditions and parameters apply.

(Subscript[t, 0]) = 0.0 s                (Subscript[θ, 0]) = (11^O)              (Subscript[ω, 0]) = 0.0 rad/s

Length of pendulum:  14.7 m (it's big)
Drag coefficient: 490 N-m
Mass of pendulum: 100 kg
Driving force amplitude: 1180 N
Angular frequency of driving force: 0.67 rad/s

Integration step size: 0.025 s

2. The Poincare Section.

In the previous laboratory we found that the phase-space trajectory of the physical pendulum is complex and difficult to decipher. We now develop a technique to view what is called the Poincare section of the phase space of the sytem.

Consider a phase-space trajectory of a dynamical system like the one we generated for the realistic physical pendulum. The system clearly fell within some well-defined limits in terms of θ and ω, but within those limits the trajectory was very complicated. The Poincare section is a 'stroboscopic' view of the phase-space trajectory that enables us to see the region of phase space the system occupies.

An essential component of the onset of chaos is the presence of the driving force. We will use the periodicity of the driving term to investigate the phase space of our physical pendulum in the following way. (1) A phase-space trajectory is developed with the same algorithms we've used before. (2) Instead of plotting every point we only plot those times that are in phase with the driving torque. In other words, we only plot those points on our phase-space trajectory where (Subscript[ω, drive]) t  =  2  nπ where n is an integer.

3. Necessary Commands.

To generate the Poincare section for the physical pendulum we need to use a different Mathematica command to determine the trajectory. The Table command we have used in the past saves ALL the points on the phase-space trajectory and we want to save only selected points. The code below demonstrates the use of a Do loop and the necessary If statements to store selected points in a table. In it we solve the following dynamical model. 

              ((dx^2)/(dt^2))  =  -x           \(TraditionalForm
      \`x(t = 0)\  = \ 
        \(1.0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(\(x\^′\)(t = 0)\)\  = \ 
          0.2\)\)
              
We will keep only those points when t=nπ.

\(\(\( (*\ 
      initial\ conditions\ and\ parameters\ *) \)\(
\)\(\(t0\  = \ 0.0;\)

    \(x0\  = \ 1.0;\)

    \(v0\  = \ 0.2;\)

    \(step\  = \ 0.01;\)

 (*\ 
      get\ the\ second\ and\ third\ points
	\ \ on\ the\ curve\ *) 


    \(t1\  = \ t0\  + \ step;\)

    \(x1\  = \ x0\  + \ step*v0;\)

    \(x2\  = \ 2*x1\  - \ x0\  - \ \((step*step*x1)\);\)

    \(v1\  = \ \((x2\  - \ x1)\)/\((2*step)\);\)

 (*\ 
      put\ the\ first\ point\ in\ the\ table\ *) 


    \(MyTable\  = \ {{x0, v0}};\)

 (*\ 
      Now\ use\ a\ Do\ loop\ and\ store\ the\ points
		\ when\ t\  = \ 
        nπ . \ 
            a\ centered\ formula\ is\ used
	\ \ to\ approximate\ the\ second\ derivative . \ 

\ \ \ parameters\ needed\ to\ test\ when\ to\ store
		\ the\ \(\(data\)\(.\)\)\ *) 


    \(TimeTest\  = \ Pi;\)

    \(PeriodCounter\  = \ 1;\)

 (*\ 
      first\ point\ of\ the\ algorithm\ *) 


    \(xminus\  = \ x0;\)

    \(xmid\  = \ x1;\)

    \(xplus\  = \ x2;\)

    \(tmin\  = \ t1\  + \ step;\)

    \(tmax\  = \ 50.0;\)


    Do[xminus\  = \ xmid; 
	\ \ \ \ \ xmid\  = \ 
        xplus; 
	\ \ \ \ \ xplus\  = \ 
        2*xmid\  - \ 
          xminus\  - \ \((step*step*
              xmid)\); 
	\ \ \ \ \ vmid\  = \ \((xplus\  - \ xminus)\)/\((2*
              step)\); 
		\ \ If[
        t > \ TimeTest, 
		\ \ \ \ \ \ \ \ \ MyTable\  = \ 
          Append[MyTable, {xmid, 
              vmid}]; 
		\ \ \ \ \ \ \ \ \ PeriodCounter\  = \ 
          PeriodCounter\  + \ 1; 
		\ \ \ \ \ \ \ \ \ TimeTest\  = \ 
          PeriodCounter*2*Pi
		\ \ \ \ \ \ \ ], 
	\ \ \ \ \ {t, tmin, 
        tmax, step}
\ \ ]\)\)\)

\(\(\( (*\ Plot\ the\ \(\(results\)\(.\)\)\ *) \)\(
\)\(ListPlot[
      MyTable, 
	PlotRange -> {{\(-1.2\), 1.2}, {\(-1.2\), 1.2}}, 	
	
      PlotJoined -> False, 
	Frame -> True, 
	
      FrameLabel -> {"x", "v", "", ""}, 
	
      PlotStyle -> {RGBColor[1, 0, 0], PointSize[0.020]}
	]\)\)\)

4. Investigating the Physics.

Once you have a working code consider the following questions.

a. What does the Poincare section look like when you turn off the torques due to friction and the driving term? The language used to describe the features you discover here is that the system is drawn to the surface defined by the Poincare section. This surface is called an attractor. What would you name the attractor for this system?

b. What does the Poincare section look like when you turn on the torque due to friction, but keep the driving term turned off?  What would you name the attractor for this system with the damping term turned on?

c. Turn everything on and generate the Poincare section. What does it look like? Compare this result with parts 4.a and 4.b. The surface you have discovered is called a strange attractor .

d. What happens to the strange attractor if you change the inital conditions slightly?

e. What happens to the strange attractor if you shift the time of the snapshot by one-fourth of the period of the driving term?

f. Numerical methods  enable us to examine systems that are much more complex than before now. In the course of studying systems like the non-linear, damped, forced, physical pendulum that you have just studied we observe an extreme sensitivity to the initial conditions. What impact, if any, does this sensitivity have on the predictive power of our models of complex systems?