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\title{Lab 2: Chaos}
\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.
\end{abstract}


\section{Chaos}
\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}


    \subsection{Model of a Driven Pendulum}
    \label{sec:modeling}
    The driven pendulum exemplifies chaotic motion. In the angular coordinate $\theta$, the equation of motion of a driven pendulum is 

    \begin{center}
        $\frac{d^2}{dt^2} = \frac{{\omega_0}^2}{I} sin(\theta) - \frac{{\alpha}}{I} \frac{d\theta}{dt} + \frac{f}{I} cos(\omega t + \phi)$.
    \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.


\section{Phase Space}
\label{sec:phasespace}
A 2-dimensional phase space is a useful environment in which to identify the chaos in the motion in a coordinate. For an oscillator, the convenient coordinate is its angular position $\theta$, with the angular velocity response $\theta^\prime$. In figure \ref{fig:model_periodic}, observe how a periodic variable can be identified in phase space.

\begin{figure}
    \includegraphics[width=6.5in]{model_periodic.png}
    \caption{Computer-generated model of a periodic function in phase space. A pendulum's angular coordinate would correspond to $\theta=x$. The motion is predictable and orbits a single point.}
    \label{fig:model_periodic}
\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}
    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.

    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}
        \label{fig:model_damped_phase}
        \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}}
    \end{figure}

    \begin{figure}
        \label{fig:model_driven_phase}
        \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.}
    \end{figure}

    \begin{figure}
        \label{model_driven_nodamp_phase}
        \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.}
        \label{fig:model_no_drag}
    \end{figure}












\section{Chaos compared against Randomness}
\label{sec:reverbmap}
A random variable has no discernable pattern in its output. If a variable has a random response as a function of time, then it will appear as a random scatter in both the time domain and in phase space. A random response to the position coordinate is plotted in figure \ref{fig:random_phase}, in phase space. The random character can be see in the fact that the distribution of first-derivative responses has no discernable pattern.

A poincar\'{e} map takes a periodic input and samples the output of a variable response in phase space. If the system is periodic with that sampling period, the poincar\'{e} map will plot only a single dot, which is the same response seen after each period, e.g. figure \ref{fig:periodic_poincare}. If the response is random, the poincar\'{e} map will have a random distribution, similar to that seen in the phase map (figure \ref{fig:random_poincare}. If the response is chaotic, we expect to see a distribution of points that is not entirely random but is not confined to a single point. This further demonstrates the continuum behaviour where a chaotic variable extends the behaviour of a periodic variable but not to the point of randomness.

The driven pendulum is a physical system that is known to exhibit chaotic behaviour. The model for this system can be expressed analytically given a small-angle approximation 



The phase map and poincar\'{e} map are generated from the model of a driven pendulum

\begin{figure}
    \includegraphics[width=6.5in]{chaotic_b_time.png}
    \caption{Computer-generated model of a driven pendulum with no damping. Arguments: $\theta=x, f=1, I=1, \omega=0.5, \alpha=0, k=\frac{\omega_0^2}{I}=1.5, \phi=0$, 180 time steps.}
    \label{fig:chaotic_b_time}
\end{figure}



\section{Discussion}
\label{sec:discussion}



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