new images

chaos/report/report.tex

@@ -103,37 +103,80 @@ 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_time}. Fourier transformations of the waveform in figure \ref{fig:exp_damped_fourier} 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 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 $0.93\pm0.6 Hz$. 
 
     \begin{figure}
-        \includegraphics[width=6.5in]{exp_damped_time.png}
-        \caption{The damped oscillations of the experimental pendulum. The fourier transforms of this function reveal the natural frequency of the pendulum, in figure \ref{fig:exp_damped_fourier}.}
-        \label{fig:exp_damped_time}
+        \hfill
+        {\includegraphics[width=6in]{exp_damped_time.png}}[A]
+        \hfill
+        {\includegraphics[width=5in]{exp_damped_fourier.png}}[B]
+        \hfill
+        \caption{[A] The damped oscillations of the experimental pendulum. The fourier transforms of this function reveal the natural frequency of the pendulum, in figure B. [B] The fourier frequency-space representation of the damped pendulum. The peak near $0.9Hz$ is the natural/resonant frequency of this pendulum.}
+        \label{fig:exp_damped}
     \end{figure}
 
 
-    \begin{figure}
-        \includegraphics[width=6.5in]{exp_damped_fourier.png}
-        \caption{The fourier frequency-space representation of the damped oscillating. The peak near 0.8 Hz is the natural/resonant frequency of this pendulum.}
-        \label{fig:exp_damped_fourier}
-    \end{figure}
-
-
-
-
     \subsection{Periodic Motion}
-    To produce periodic motion, the driving arm is run with a driving frequency $1.45 \pm 0.02s$. The observed motions are plotted in phase space in figure \ref{fig:exp_periodic_phase}. The motion still suggests some chaotic behaviour by the variation in the path taken about the critical point, but the poincar\'e section in figure \ref{fig:exp_periodic_poincare} shows that the orbit is converging.
+    To produce periodic motion, the driving arm is run with a driving period $1.45\pm0.02s$. The observed motion is plotted in figure \ref{fig:exp_periodic}. The motion still suggests the possibility of chaos by the variation in the path taken about the critical point, but the poincar\'e section in figure \ref{fig:exp_periodic} indicates that the orbit is likely converging. The period of the observed motion is consistent with the driving period.
 
     \begin{figure}
-        \includegraphics[width=6.5in]{exp_periodic_phase.png}
-        \caption{The periodic motion at driving period $1.45 s$is easily recognizable in phase space. It follows a stable orbit around the attractor. This plot still suggests chaotic behaviour, as the }
-        \label{fig:exp_periodic_phase}
+        \hfill
+        [A]{
+            \includegraphics[width=1.8in]{exp_periodic_time.png}
+            }
+        \hfill
+        [B]{
+            \includegraphics[width=1.8in]{exp_periodic_phase.png}
+            }
+        \hfill
+        [C]{
+            \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.}
     \end{figure}
 
+
+    \subsection{Chaotic Motion}
+    The driving arm period was increased until the pendulum was exhibiting visibly complex behaviour. The final period was $1.13\pm0.02s$. In figure \ref{fig:exp_chaotic_A} the observed motions are plotted. Two chaotic attractors are strikingly visible in the phase diagram, and the poincar\'e plot shows significant deviation through the phase. 
+
     \begin{figure}
-        \includegraphics[width=6.5in]{exp_periodic_poincare.png}
-        \caption{The periodic motion at driving period $1.45 s$is easily recognizable in phase space. It follows a stable orbit around the attractor. This plot still suggests chaotic behaviour, as the }
-        \label{fig:exp_periodic_poincare}
+        \hfill
+        [A]{
+            \includegraphics[width=1.8in]{exp_chaotic_A_time.png}
+            }
+        \hfill
+        [B]{
+            \includegraphics[width=1.8in]{exp_chaotic_A_phase.png}
+            }
+        \hfill
+        [C]{
+            \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.}
+    \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.
+
+    \begin{figure}
+        \hfill
+        [A]{
+            \includegraphics[width=1.8in]{exp_chaotic_B_time.png}
+            }
+        \hfill
+        [B]{
+            \includegraphics[width=1.8in]{exp_chaotic_B_phase.png}
+            }
+        \hfill
+        [C]{
+            \includegraphics[width=1.8in]{exp_chaotic_B_poincare.png}
+            }
+        \hfill
+        \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}