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@ -57,39 +57,35 @@ A 2-dimensional phase space is a useful environment in which to identify the cha
\label{fig:model_periodic} \label{fig:model_periodic}
\end{figure} \end{figure}
\subsection{Poincar\'{e} Map}
A Poincar\'{e} section is a 2D phase space cross-section, in this case corresponding to the repetition of a certain period, such as the natural or driving period of a forced pendulum. One draws a map between each successive point to create a Poincar\'e map, which is a useful representation of a system's behaviour in phase space.
An undriven and undamped oscillator will always return to the same point after one period, so a Poincar\'e map sampled using the period corresponding to the oscillator's natural frequency will consist of a single dot.
\subsection{Chaotic Attractor} \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 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}.
Damping still plays an important role in this chaotic motion; figure \ref{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. 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.
Computer models were generated using the Chaos for Java program written by Brian Davies. \cite{CHAOSFORJAVA} The path is computed from the circle to the triangle. Computer models were generated using the Chaos for Java program written by Brian Davies. \cite{CHAOSFORJAVA} The path is computed from the circle to the triangle.
\begin{figure} \begin{figure}
\label{fig:model_damped_phase}
\includegraphics[width=6.5in]{model_damped_phase.png} \includegraphics[width=6.5in]{model_damped_phase.png}
\caption{Model of a damped oscillator in phase space. The critical points are stable, since the velocity approaches zero from all points within the associated region. \cite{CHAOSDYNAMICS}} \caption{Model of a damped oscillator in phase space. The critical points are stable, since the velocity approaches zero from all points within the associated region. \cite{CHAOSDYNAMICS}}
\label{fig:model_damped_phase}
\end{figure} \end{figure}
\begin{figure} \begin{figure}
\label{fig:model_driven_phase}
\includegraphics[width=6.5in]{model_driven_phase.png} \includegraphics[width=6.5in]{model_driven_phase.png}
\caption{Computer-generated model of a damped and driven pendulum's angular motion. The motion jumps between critical points (called attractors in this context), and exhibits what seem to be two semi-stable energy states. Arguments: $\theta=x, f=1, I=1, \omega=2/3, \alpha=0.7, k=\frac{\omega_0^2}{I}=2, \phi=0, \theta(0)=1, \theta^\prime(0)=1$, 150 time steps.} \caption{Computer-generated model of a damped and driven pendulum's angular motion. The motion jumps between critical points (called attractors in this context), and exhibits a semi-stable low-energy state around one of attractors. Arguments: $\theta=x, f=1, I=1, \omega=2/3, \alpha=0.7, k=\frac{\omega_0^2}{I}=2, \phi=0, \theta(0)=1, \theta^\prime(0)=1$, 150 time steps.}
\label{fig:model_driven_phase}
\end{figure} \end{figure}
\begin{figure} \begin{figure}
\label{model_driven_nodamp_phase}
\includegraphics[width=6.5in]{no_drag.png} \includegraphics[width=6.5in]{no_drag.png}
\caption{Computer-generated model of a driven pendulum with no damping. The path through phase space is unstable and could easily take off toward either extreme of the $\theta$ coordinate. This sensitivity is characteristic of chaotic motion. Arguments: $\theta=x, f=1, I=1, \omega=0.5, \alpha=0, k=1.5, \phi=0$, 180 time steps.} \caption{Computer-generated model of a driven pendulum with no damping. The path through phase space is unstable and could easily take off toward either extreme of the $\theta$ coordinate. This sensitivity is characteristic of chaotic motion. Arguments: $\theta=x, f=1, I=1, \omega=0.5, \alpha=0, k=1.5, \phi=0$, 180 time steps.}
\label{fig:model_no_drag} \label{fig:model_driven_nodamp_phase}
\end{figure} \end{figure}
\subsection{Poincar\'{e} Map}
A Poincar\'{e} section is a 2D phase space cross-section, in this case corresponding to the repetition of a certain period, such as the natural or driving period of a forced pendulum. One draws a map between each successive point to create a Poincar\'e map, which is a useful representation of a system's behaviour in phase space.
An undriven and undamped oscillator will always return to the same point after one period, so a Poincar\'e map sampled using the period corresponding to the oscillator's natural frequency will consist of a single dot.