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As Strogatz says in reference [1], ``No definition of the term chaos is universally accepted yet, but almost everyone would agree on the three ingredients used in the following working definition''.
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As Strogatz says in reference [1], "No definition of the term chaos is universally accepted yet, but almost everyone would agree on the three ingredients used in the following working definition".
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| Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions. |
Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions. |
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| Aperiodic long-term behavior means that there are trajectories which do not settle down to fixed points, periodic orbits, or quasiperiodic orbits as $t \to \infty$. For the purposes of this definition, a trajectory which approaches a limit of $\infty$ as $t \to \infty$ should be considered to have a fixpoint at $\infty$. |
Aperiodic long-term behavior means that there are trajectories which do not settle down to fixed points, periodic orbits, or quasiperiodic orbits as $t \to \infty$. For the purposes of this definition, a trajectory which approaches a limit of $\infty$ as $t \to \infty$ should be considered to have a fixpoint at $\infty$. |
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| Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast, i.e., the system has a positive Liapunov exponent. |
Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast, i.e., the system has a positive Liapunov exponent. |
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| Strogatz notes that he favors additional contraints on the aperidodic long-term behavior, but leaves open what form they may take. He suggests two alternatives to fulfill this: |
Strogatz notes that he favors additional contraints on the aperidodic long-term behavior, but leaves open what form they may take. He suggests two alternatives to fulfill this: |
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| \begin{enumerate} |
\begin{enumerate} |
| \item Requiring that $\exists$ an open set of initial conditions having aperiodic trajectories, or |
\item Requiring that $\exists$ an open set of initial conditions having aperiodic trajectories, or |
| \item If one picks a random initial condition $x(0)$ then there must be a nonzero chance of the associated trajectory $x(t)$ being aperiodic. |
\item If one picks a random initial condition $x(0)$ then there must be a nonzero chance of the associated trajectory $x(t)$ being aperiodic. |
| \end{enumerate} |
\end{enumerate} |
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| \subsection{Further reading} |
\subsection{Further reading} |
| \begin{enumerate} |
\begin{enumerate} |
| \item B. Codenotti and Luciano Margara. Chaos in Mathematics, Physics, and Computer Science: Similarities and Dissimilarities. http://pespmc1.vub.ac.be/Einmag\_Abstr/BCodenotti.html |
\item B. Codenotti and Luciano Margara. Chaos in Mathematics, Physics, and Computer Science: Similarities and Dissimilarities. http://pespmc1.vub.ac.be/Einmag\_Abstr/BCodenotti.html |
| \end{enumerate} |
\end{enumerate} |
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| \subsection{References} |
\subsection{References} |
| \begin{enumerate} |
\begin{enumerate} |
| \item Steven H. Strogatz, "Nonlinear Dynamics and Chaos". Westview Press, 1994. |
\item Steven H. Strogatz, "Nonlinear Dynamics and Chaos". Westview Press, 1994. |
| \end{enumerate} |
\end{enumerate} |
