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\begin{document}
\raggedright

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%Slide 1
\title{{\LARGE\bf Nonlinear Systems}\newline
              (\ldots and chaos) \newline
                a brief introduction
\newline
}
\author{
\begin{wrapfigure}{r}{0.35\textwidth}
  \vspace{-2cm}
  \begin{center}
    {\fontsize{5}{5}\selectfont  
       http://csustan.csustan.edu/\~\ tom
    }
   %% \includegraphics[width=0.15\textwidth]{tom-carter-QR.png}
   {\hyperref{http://csustan.csustan.edu/\~tom}{}{}
     {    \includegraphics[width=0.2\textwidth]{tom-carter-QR-pic.png}
     }
   }
  \end{center}
  \end{wrapfigure}
      Tom Carter \newline
      \newline
     Computer Science\newline
     CSU Stanislaus
%Computer Science\newline
%CSU Stanislaus\newline
\vfill
{\hyperref{http://csustan.csustan.edu/\~tom/Lecture-Notes/Nonlinear-Systems/Nonlinear-Systems.pdf}{}{}
     {http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Nonlinear-Systems/Nonlinear-Systems.pdf}}
}


%\date{Fall, 2011}
\date{\today}


\maketitle

%Slide 2
%% \sectionhead{Our general topics:}
%% %%tth:\begin{itemize}
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%% 	        {$\circledcirc$ References\newline}} 
%% %%tth:\makehyperlink{References}
%% %%tth:\end{itemize}
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%% 
%% \pagedone

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\pagedone




\sectionhead{What are nonlinear systems?}

\begin{itemize}
       \item Let's start by adding another word to this, and ask the question, "What are nonlinear dynamical systems?" (and we'll go
         back to front \ldots)
         
         A reasonable thing to say is that a {\em system} is a collection of entities that we can treat (for some purpose, in
         some context) as a unity of interacting parts or elements.  At various times we will treat various collections of
         entities as systems, or subsystems.  We may at times ignore certain elements that might otherwise be included.
         There will also often be times when we will engage in abstraction, and refer to a ``system'' when we are actually discussing,
         manipulating, or analyzing an abstraction from the real (physical) system we are interested in.
         
         A {\em dynamical} system is one which changes over time.  It is generally not unreasonable to assume that there is
         some form of energy flow involved in such a system.
         
         A {\em nonlinear} dynamical system is, as the name implies, a system whose best description (behavior) is not
         {\em linear}.  There are various contexts and forms of description for the concept {\em linear}.  We'll look at a variety
         of examples as we go along.

 \pagedone
      \end{itemize}
      
 \include{Linear-String}
        
\include{Discrete-Linear}

\include{Discrete-Nonlinear}

\include{Other-Topics}

\pagedone
\sectionhead{Appendix 1}        
 
          Just to check (and practice our derivatives :-) with the damped harmonic oscillator example \ldots
          
          We had:
          \begin{align*}
               x(t) &= e^{-\frac{b}{2}t}*2*\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)}
          \end{align*}
          from which,
          \begin{align*}
               v(t) &=  \dot{x}(t) = -\frac{b}{2}e^{-\frac{b}{2}t}*2\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)} \\
                              &   \ \ \ \ -e^{-\frac{b}{2}t}*\sqrt{4-b^2}\sin{\left(\frac{\sqrt{4-b^2}}{2}t\right)}\\
                              & = -e^{-\frac{b}{2}t}\left(b*\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)}
                                                    + \sqrt{4-b^2}*\sin{\left(\frac{\sqrt{4-b^2}}{2}t\right)}\right)
          \end{align*}
\pagedone
          Combining the terms:
          \begin{align*}
               & -x(t) - bv(t) \\
                               & = - e^{-\frac{b}{2}t}*2\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)} \\
                               &  \ \ \ + be^{-\frac{b}{2}t}\left(b*\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)}
                                                    + \sqrt{4-b^2}\sin{\left(\frac{\sqrt{4-b^2}}{2}t\right)}\right) \\
                               &=  e^{-\frac{b}{2}t}\left((b^2-2)\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)}
                                                    + b\sqrt{4-b^2}\sin{\left(\frac{\sqrt{4-b^2}}{2}t\right)}\right)
          \end{align*}
\pagedone
          And, sure enough, when we calculate $\dot{v}(t)$, we get the same thing:          
          \begin{align*}
               \dot{v}(t) &=  \frac{b}{2}e^{-\frac{b}{2}t}\left(b*\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)}
                                                    + \sqrt{4-b^2}\sin{\left(\frac{\sqrt{4-b^2}}{2}t\right)}\right) \\
                              &   \ \ \ \ - e^{-\frac{b}{2}t}\left(-b*\frac{\sqrt{4-b^2}}{2}\sin{\left(\frac{\sqrt{4-b^2}}{2}t\right)}
                                                + \frac{4-b^2}{2}\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)}\right)\\
                              & = e^{-\frac{b}{2}t}\left(\frac{b^2}{2}\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)}
                                                    + b\frac{\sqrt{4-b^2}}{4}\sin{\left(\frac{\sqrt{4-b^2}}{2}t\right)}\right) \\
                              &   \ \ \ \ +e^{-\frac{b}{2}t}\left(b\frac{\sqrt{4-b^2}}{2}\sin{\left(\frac{\sqrt{4-b^2}}{2}t\right)}
                                                - \frac{4-b^2}{2}\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)}\right)\\
                              & = e^{-\frac{b}{2}t}\left((b^2 - 2)\cos{\left(\frac{\sqrt{4-b^2}}{2}t\right)}
                                      +b\sqrt{4-b^2}\sin{\left(\frac{\sqrt{4-b^2}}{2}t\right)}\right)\\
          \end{align*}
\pagedone

\sectionhead{Appendix 2}
        Something will go here \ldots
                            
\pagedone
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 \footnotesize
 \bibliographystyle{plain} 
 


\marginpar{\hyperlink{Contents}{\hfil$\leftarrow$}}

\vspace{-\baselineskip}
\vspace{-\baselineskip}
\vspace{-\baselineskip}
 
 \begin{thebibliography}{12}

\addcontentsline{toc}{section}{References}

\bibitem{banks}
Banks, J., Brooks, J., Cairns, G., Davis, G., and Stacey, P.,\newline
{\em On Devaney's Definition of Chaos},\newline
American Mathematical Monthly, Vol. 99, No. 4, Apr. 1992, Mathematical Association of America.\newline
\hyperref{http://www.jstor.org/pss/2324899}{}{}
{http://www.jstor.org/pss/2324899}

\bibitem{baryam}
Bar-Yam, Yaneer,\newline
{\em Dynamics of Complex Systems (Studies in Nonlinearity) },
Westview Press, Boulder, 1997.

\bibitem{bhaumik}
Bhaumik,Indranil, and Choudhury, Binayak S.\newline
{\em The Shift Map and the Symbolic Dynamics and Application of Topological Conjugacy}\newline
Journal of Physical Sciences, Vol. 13, 2009, 149-160 ISSN: 0972-8791.
\hyperref{http://www.vidyasagar.ac.in/journal/}{}{}
{http://www.vidyasagar.ac.in/journal/}\newline
\hyperref{http://vidyasagar.ac.in/journal/maths/vol13/art13.pdf}{}{}
{http://vidyasagar.ac.in/journal/maths/vol13/art13.pdf}

\bibitem{bishop}
Bishop, Robert,\newline
 "Chaos", The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.).\newline
 \hyperref{http://plato.stanford.edu/archives/fall2009/entries/chaos/}{}{}
{http://plato.stanford.edu/archives/fall2009/entries/chaos/}

\bibitem{briggs}
Briggs, Keith Martin\newline
{\em Feigenbaum Scaling in Discrete Dynamical Systems}\newline
Ph.D. Thesis to University of Melbourne, Australia\newline
\hyperref{http://keithbriggs.info/documents/Keith_Briggs_PhD.pdf}{}{}
{http://keithbriggs.info/documents/Keith\_Briggs\_PhD.pdf}

\bibitem{cambel}
\c{C}ambel, Ali Bulent,\newline
{\em Applied Chaos Theory: A Paradigm for Complexity}\newline
Academic Press, 1992.

\bibitem{carter1}
Carter, Thomas,\newline
{\em Nonlinear Systems (\ldots and chaos) a brief introduction}\newline
(this document, with LaTeX source, etc.)\newline
\hyperref{http://csustan.csustan.edu/~tom/Lecture-Notes/Nonlinear-Systems/}{}{}
{http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Nonlinear-Systems/}

\bibitem{carter2}
Carter, Thomas,\newline
{\em What Shape is a Circle?}\newline
\hyperref{http://csustan.csustan.edu/~tom/Lecture-Notes/Circles/Circles.pdf}{}{}
{http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Circles/Circles.pdf}

\bibitem{carter3}
Carter, Thomas,\newline
{\em Entropy, Power Laws, and Economics}\newline
\hyperref{http://csustan.csustan.edu/~tom/Lecture-Notes/Econ101/econ101.pdf}{}{}
{http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Econ101/econ101.pdf}

\bibitem{carter4}
Carter, Thomas,\newline
{\em Some Fractals and Fractal Dimensions}\newline
\hyperref{http://csustan.csustan.edu/~tom/Lecture-Notes/Fractal-Dimension/Fractal-dimension.pdf}{}{}
{http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Fractal-Dimension/Fractal-dimension.pdf}

\bibitem{carter5}
Carter, Thomas,\newline
{\em An introduction to information theory and entropy}\newline
\hyperref{http://csustan.csustan.edu/~tom/Lecture-Notes/Information-Theory/info-lec.pdf}{}{}
{http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Information-Theory/info-lec.pdf}

\bibitem{carter6}
Carter, Thomas,\newline
{\em A brief survey of linear algebra}\newline
\hyperref{http://csustan.csustan.edu/~tom/Lecture-Notes/Linear-Algebra/lin-alg.pdf}{}{}
{http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Linear-Algebra/lin-alg.pdf}

\bibitem{carter7}
Carter, Thomas,\newline
{\em The Logistic Flow (Continuous)}\newline
\hyperref{http://csustan.csustan.edu/~tom/Lecture-Notes/Logistic/Logistic-continuous.pdf}{}{}
{http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Logistic/Logistic-continuous.pdf}

\bibitem{carter8}
Carter, Thomas,\newline
{\em Peripatetic (A random walk in random walks)}\newline
\hyperref{http://csustan.csustan.edu/~tom/Lecture-Notes/Random-Walk/peripatetic.pdf}{}{}
{http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Random-Walk/peripatetic.pdf}

\bibitem{carter9}
Carter, Thomas,\newline
{\em Symbolic Dynamics (a brief introduction)}\newline
\hyperref{http://csustan.csustan.edu/~tom/Lecture-Notes/Symbolic-Dynamics/Symbolic-Dynamics.pdf}{}{}
{http://csustan.csustan.edu/\~\ tom/Lecture-Notes/Symbolic-Dynamics/Symbolic-Dynamics.pdf}

\bibitem{davies}
Davies, Brian,\newline
Chaos Explorer\newline
\hyperref{http://maths.anu.edu.au/\~briand/chaos/software-applet.html}{}{}
{http://maths.anu.edu.au/\~\ briand/chaos/software-applet.html}

\bibitem{devaney}
Devaney, Robert,\newline
{\em An Introduction to Chaotic Dynamical Systems, 2nd Edition}\newline
Westview Press, Boulder, 2003.

\bibitem{ekeland}
Ekeland, Ivar,\newline
{\em Mathematics and the Unexpected}\newline
University of Chicago Press, 1990.

\bibitem{feigenbaum}
Feigenbaum, Mitchell,\newline
{\em Universal behavior in nonlinear systems}\newline
Los Alamos Science, Summer 1980, 1(1):4-27,\newline
Los Alamos National Laboratory Library, 1980.\newline
 \hyperref{http://library.lanl.gov/cgi-bin/getfile?01-01.pdf}{}{}
{http://library.lanl.gov/cgi-bin/getfile?01-01.pdf}
 
 \bibitem{fillion}
 Fillion, Nicolas,\newline
 {\em Order in Chaos: Universal Behavior in Nonlinear Systems}\newline
 Final project for
AM9570B: Introduction to Mathematical Modelling and Simulation,
Presented to Professor Pei Yu\newline
 Department of Philosophy\newline
 The University of Western Ontario, April 21, 2010\newline
 \hyperref{http://www.nfillion.com/docs/project.pdf}{}{}
{http://www.nfillion.com/docs/project.pdf}
 
 \bibitem{girvan}
 Girvan, Ray, \newline
 {\em The Mandelbrot Monk}\newline
 Web publication -- 1999/04/01\newline
  \hyperref{ http://classes.yale.edu/fractals/Mandelset/MandelMonk/MandelMonk.html}{}{}
{ http://classes.yale.edu/fractals/Mandelset/MandelMonk/MandelMonk.html}
 \hyperref{http://csustan.csustan.edu/~tom/SFI-CSSS/misc/14\%20Caution\%20(Do\%20Not\%20Stop\%20On\%20Tracks).mp3}{}{}
{Caution: Do not stop on the tracks} (Furthur - 2011/10/07)


 \bibitem{gleick}
 Gleick, James,\newline
 {\em Chaos: Making a New Science},\newline
 Penguin (Non-Classics); Anv Rep edition, 2008.
 
 \bibitem{knudsen}
 Knudsen, Carsten \newline
 {\em Chaos Without Nonperiodicity}\newline
 American Mathematical Monthly, Vol. 101, No. 6, Jun. - Jul. 1994,
 Mathematical Association of America. \newline
 \hyperref{http://www.jstor.org/stable/2975324}{}{}
{http://www.jstor.org/stable/2975324}
 
 \bibitem{li}
 Li, Tien-Yien, and Yorke, James A.\newline
 {\em Period Three Implies Chaos}\newline
 The American Mathematical Monthly, Vol. 82, No. 10 (Dec., 1975), pp. 985-992 \newline
 Mathematical Association of America\newline
 \hyperref{http://www.jstor.org/stable/2318254}{}{}
{http://www.jstor.org/stable/2318254}
 
 \bibitem{schroeder}
Schroeder, Manfred,\newline
{\em Fractals, Chaos, Power Laws, Minutes from an Infinite Paradise},
W. H. Freeman, New York, 1991.

\bibitem{strogatz}
Strogatz, Steven, \newline
{\em Nonlinear Dynamics And Chaos: With Applications To Physics, Biology,
Chemistry, And Engineering (Studies in Nonlinearity) }\newline
Westview Press, Boulder, 2001.

\bibitem{zhan}
Zhan, Mei-Qin,\newline
Course notes, on Systems of First Order Differential Equations\newline
 \hyperref{http://www.unf.edu/\~mzhan/chapter4.pdf}{}{}
 {http://www.unf.edu/\~\ mzhan/chapter4.pdf}



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