update

chaos/report/report.tex

@@ -20,12 +20,12 @@
 \begin{document}
 
 
-\title{Lab 2: Chaos}
+\title{Lab 2: Chaos in a Driven Pendulum}
 \author{Otho Ulrich, Eugene Kopf}
 \maketitle
 
 \begin{abstract}
-The chaotic behaviour of a driven pendulum is explored. Phase space behaviours of a random response, non-chaotic periodic response, and chaotic response are generated from computational models. The functional form of each is compared, demonstrating the strange attractor as a distinguishing feature of a chaotic response. The motion of a physical pendulum is observed under damped and damped-driven conditions. The motion is characterized according to its phase space output, and determined to be chaotic.
+The chaotic behaviour of a driven pendulum is explored. Phase space behaviours of a non-chaotic periodic response and chaotic response are generated from computational models. The form of each is compared, demonstrating the chaotic attractor as a distinguishing feature of chaos. The motion of a physical pendulum is observed under damped and damped-driven conditions. The motion is explored according to its phase space output, and determined to be chaotic.
 \end{abstract}
 
 
@@ -33,7 +33,7 @@ The chaotic behaviour of a driven pendulum is explored. Phase space behaviours o
 \label{sec:chaos}
 Chaos is observed in many non-linear physical systems. It is the condition that a system's outcome is strongly sensitive to initial conditions. The changing conditions as the system evolves affect the outcome such that predicting the future state becomes impossible. The motion of a driven oscillator, such as a driven pendulum, for example, becomes unpredictable as the driving frequency and natural frequency of the pendulum interact. Damping can constrain the motion, and we find that while the motion is unpredictable, it still displays certain characteristics that can be analyzed.
 
-For a finite period of time, chaotic behaviour isn't completely discernable from periodic behaviour, because the possibility exists that the function may repeat itself at some future time. To identify chaos, one makes a judgment after enough time has elapsed to assume for practical purposes the function will not repeat. Distinctions are seen between a random variable, periodic variable, and chaotic variable in phase space and poincar\'{e} sections. \cite{TANGLEDTALE}
+For a finite period of time, chaotic behaviour isn't completely discernable from periodic behaviour, because the possibility exists that the function may repeat itself at some future time. To identify chaos, one makes a judgment after enough time has elapsed to assume for practical purposes the function will not repeat. Distinctions are seen between a periodic variable and a chaotic variable in phase space and poincar\'{e} sections. \cite{TANGLEDTALE}
 
 
     \subsection{Model of a Driven Pendulum}
@@ -41,7 +41,7 @@ For a finite period of time, chaotic behaviour isn't completely discernable from
     The driven pendulum exemplifies chaotic motion. In the angular coordinate $\theta$, the equation of motion of a driven pendulum is 
 
     \begin{center}
-        $\frac{d^2}{dt^2} = \frac{{\omega_0}^2}{I} sin(\theta) - \frac{{\alpha}}{I} \frac{d\theta}{dt} + \frac{f}{I} cos(\omega t + \phi)$.
+        $\frac{d^2\theta}{dt^2} = \frac{{\omega_0}^2}{I} sin(\theta) - \frac{{\alpha}}{I} \frac{d\theta}{dt} + \frac{f}{I} cos(\omega t + \phi)$.
     \end{center}
 
     Here, $\omega_0$ represents the natural frequency of the pedulum, also its resonant frequency. The system will respond most strongly to the driver at this frequency. $\alpha$ is a damping term -- this can take a variety of forms, and in the experiment of section \ref{sec:experiment} is produced by a neodymium magnet interacting with the metal wheel of the pendulum. $f$ is the forcing amplitude where $\omega$ is the forcing frequency, offset from the angular coordinate by a phase $\phi$. $I$ is the moment of inertia of the pendulum.
@@ -58,7 +58,7 @@ A 2-dimensional phase space is a useful environment in which to identify the cha
 \end{figure}
 
     \subsection{Chaotic Attractor}
-    Figure \ref{fig:model_damped_phase} demonstrates a damped oscillator, which exhibits stable critical points where the velocity goes to zero. Under forced conditions, the velocity does not converge to zero, but the motion produces orbits about these critical points in phase space, and we call these points chaotic attractors. Poincar\'{e} first postulated that chaos would be exemplified by complicated paths that roughly follow one of these orbits (the apex) about these attractors. \cite{CHAOSAT50} Attractors are a primary identifying characteristic of chaos, and should be observable in the chaotic motion of a forced pendulum. \cite{CHAOSDYNAMICS} A driven oscillator's path orbits around these critical points but can be seen to jump between them in an unpredictable way along the position coordinate; observe figure \ref{fig:model_driven_phase}.
+    Figure \ref{fig:model_damped_phase} demonstrates a damped oscillator, which exhibits stable critical points where the angular velocity goes to zero. Under forced conditions, the angular velocity does not converge to zero, but the motion produces orbits about these critical points in phase space, and we call these points chaotic attractors. Poincar\'{e} first postulated that chaos would be exemplified by complicated paths that roughly follow one of these orbits (the apex) about these attractors. \cite{CHAOSAT50} Attractors are a primary identifying characteristic of chaos, and should be observable in the chaotic motion of a forced pendulum. \cite{CHAOSDYNAMICS} A driven oscillator's path orbits around these critical points but can be seen to jump between them in an unpredictable way along the position coordinate; observe figure \ref{fig:model_driven_phase}.
 
     Damping still plays an important role in this chaotic motion; figure \ref{fig:model_driven_nodamp_phase} shows a driven pendulum with no damping, where the motion freely exhibits the effects of both the natural and forcing frequency, and becomes extremely complicated. The natural (un-driven) response of the pendulum is seen in the tall cirular strokes, which represent the pendulum's weight attempting to bring the pendulum to equilibrium. The driving frequency produces small-amplitude variations when the pendulum has a high angular speed and when the pendulum has a low angular speed, it can easily reverse the motion of the pendulum. The oscillator jumps around to many possible states, making it difficult to discern attractors. Chaotic attractors are much more clear when forcing interacts with damping in a system; figure \ref{fig:model_driven_phase} shows this very well.
 
@@ -103,7 +103,7 @@ An experimental driven pendulum is built in order to determine whether its motio
 \end{figure}
 
     \subsection{Resonant Frequency}
-    To determine the natural frequency of the pendulum, it was released from the top position under damped conditions and its frequency of oscillation was measured. The waveform is shown in figure \ref{fig:exp_damped}. Fourier transformations of the waveform in figure \ref{fig:exp_damped} reveal the natural frequency as the frequency where the power spectral density peaks, discounting the low-frequency peak from the flattened waveform due to damping. The natural frequency is estimated to be $0.93\pm0.6 Hz$. 
+    To determine the natural frequency of the pendulum, it was released from the top position under damped conditions and its frequency of oscillation was measured. The waveform is shown in figure \ref{fig:exp_damped}. Fourier transformations of the waveform in figure \ref{fig:exp_damped} reveal the natural frequency at where the power spectral density peaks, discounting the low-frequency peak from the flattened waveform due to damping. The natural frequency is estimated to be $0.93\pm0.6 Hz$. 
 
     \begin{figure}
         \hfill
@@ -133,8 +133,8 @@ An experimental driven pendulum is built in order to determine whether its motio
             \includegraphics[width=1.8in]{exp_periodic_poincare.png}
             }
         \hfill
-        \label{fig:exp_periodic}
         \caption{[A] The observed motion when the driving arm is run with period $1.45\pm0.02s$ is periodic at $1.48\pm0.08s$. [B] The periodic nature is recognizable in phase space. [C] The poincar\'e section shows the orbit is likely converging.}
+        \label{fig:exp_periodic}
     \end{figure}
 
 
@@ -155,8 +155,8 @@ An experimental driven pendulum is built in order to determine whether its motio
             \includegraphics[width=1.8in]{exp_chaotic_A_poincare.png}
             }
         \hfill
-        \label{fig:exp_chaotic_A}
         \caption{[A] The observed motion when the driving arm is run with period $1.13\pm0.02s$ is chaotic. [B] Chaotic attractors are recognizable in phase space. [C] The poincar\'e section shows chaos in the orbit, but that the motions are still constrained to a finite area in phase space.}
+        \label{fig:exp_chaotic_A}
     \end{figure}
 
     The natural frequency of the pendulum is tested next. The driving period corresponding to the natural frequency was chosen as $1.26\pm0.02s$. This driving frequency found a stronger balance between the two attractors observed at the previous frequency. Figure \ref{fig:exp_chaotic_B} shows plots of the motion. While performing this test, it became clear that the driving arm was near the resonance frequency, as the apparatus began to shake itself to tipping.
@@ -175,22 +175,28 @@ An experimental driven pendulum is built in order to determine whether its motio
             \includegraphics[width=1.8in]{exp_chaotic_B_poincare.png}
             }
         \hfill
+        \caption{[A] The observed motion when the driving arm is run with period $1.26\pm0.02s$ (near the resonance frequency) is chaotic. [B] . [C] The poincar\'e section shows the same integrated area of constraint in variation as the first test, but more evenly distributed.}
         \label{fig:exp_chaotic_B}
-        \caption{[A] The observed motion when the driving arm is run with period $1.26\pm0.02s$ (near the resonance frequency) is chaotic. [B] . [C] The poincar\'e section shows more variation than in the first test, but that the motions are still constrained by the damping.}
     \end{figure}
 
 
-\begin{figure}
-    \includegraphics[width=6.5in]{chaotic_b_time.png}
-    \caption{Driven oscillating with chaotic motion.}
-    \label{fig:chaotic_b_time}
-\end{figure}
-
+    One final driving period was chosen for good measure. At $1.15\pm0.02s$, the motion appears closer to sinsuisoidal than in the pervious tests. Nonetheless, the attractors can be observed in the phase diagram and the paths through the poincar\'e section are limited to the same integrated area. 
 
+    \begin{figure}
+        \hfill
+        \includegraphics[width=1.8in]{exp_chaotic_C_time.png}
+        \hfill
+        \includegraphics[width=1.8in]{exp_chaotic_C_phase.png}
+        \hfill
+        \includegraphics[width=1.8in]{exp_chaotic_C_poincare.png}
+        \hfill
+        \caption{The observed motion when the driving arm is run with period $1.16\pm0.02s$. The time graph looks significiantly different from the previous tests, but the same attractors and poincar\'e section are observed.}
+        \label{fig:exp_chaotic_C}
+    \end{figure}
 
 \section{Discussion}
 \label{sec:discussion}
-
+The periodic motion is observable in the first experimental case, plotted in figure \ref{fig:exp_periodic}. Once the driving arm period was increased from that frequency to one that induced chaotic motion, the predicted spread of intercepts through the poincar\'e section and the orbits about the chaotic attractors were visually confirmed. It would have been useful to produce the phase diagram of the damped, undriven pendulum to compare the locations of those critical points to the chaotic attractors. However, as Poincar\'e predicted so long ago, it is quite evident that constrained chaotic motion does follow complex paths about attractors, and this motion is evident in the observed motion of the damped, driven pendulum.
 
 
 \printbibliography