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chaos/report/report.tex
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@ -83,30 +83,18 @@ A 2-dimensional phase space is a useful environment in which to identify the cha
\end{figure} \end{figure}
\subsection{Poincar\'{e} Map} \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. A Poincar\'{e} section is a 2D phase space cross-section; in the case of the driven pendulum, the cross-section at a phase $\phi$ from the forcing term. 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. A poincar\'e section of the driven pendulum model is shown in figure \ref{fig:model_driven_poincare}. 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. The existence of multiple points at the same phase indicates chaotic motion.
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. \begin{figure}
\includegraphics[width=6.5in]{model_driven_poincare.png}
\caption{Poincar\'e section of the damped and driven pendulum from figure \ref{fig:model_driven_phase} at $\phi=0$. Several dots are observed at this sampling phase, so the motion can be deemed chaotic.}
\label{fig:model_driven_poincare}
\end{figure}
\section{}
\section{Chaos compared against Randomness}
\label{sec:reverbmap} \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} \begin{figure}
\includegraphics[width=6.5in]{chaotic_b_time.png} \includegraphics[width=6.5in]{chaotic_b_time.png}