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\centerline{\large \bf Interdisciplinary Nonlinear Dynamics (438)}
\centerline{Fall 2001}
\centerline{Hermann Riecke}
\vspace{.5cm}
 
\centerline{Problem Set 8}

\vspace{.5cm}

\centerline{For Discussion Section December 7}
 
\vspace{.5cm}
  
\begin{enumerate}
 
\item {\bf Poincar\'e Section and Return Map for R\"ossler System}

Consider the R\"ossler system (cf. Strogatz, ch. 12.3)
\bea
\frac{dx}{dt}&=&-y-z,\label{e:r1}\\
\frac{dy}{dt}&=&x+ay,\label{e:r2}\\
\frac{dz}{dt}&=&b+z(x-c)\label{e:r3}.
\eea
Write a Matlab program to solve (\ref{e:r1},\ref{e:r2},\ref{e:r3}) 
numerically  (using simply forward Euler).
\begin{enumerate}
\item Pick $a=0.2$ and $b=0.2$. Scan $c$ through the range $5 < c < 7.5$ and plot the 
projection of the attractor on the $(x,y)$-plane. To get a reasonable
impression of the attractor you  have to run at least to $t=500$ for
each value of the parameters.
\item Obtain the Poincar\'e section (surface of section) 
of the attractor in the $(y,z)$-plane for increasing values of $x$, i.e.
plot the points $(y_n,z_n)$ at which the attractor pierces the
$(y,z)$-plane with  increasing $x$. Thus, whenever $x$ changes
sign (with $x$ increasing)  determine $y$ and $z$ by linear
interpolation for the time at which $x=0$. Plot this discrete sequence
of $y_n$ and $z_n$ as a function of $n$. \label{s:sec}
\item Since the points on the Poincar\'e section obtained in \ref{s:sec}
lie on a line which can be characterized by   taking only the
$y_n$-values (or the $z_n$-values), obtain the return map
$y_{n+1}=f(y_n)$ by plotting the points $(y_n,y_{n+1})$. The return
map is similar in spirit to the Lorenz map for the maxima discussed in
class. Plot the function $f$ for increasing values of $c$ (with
$a=b=0.2$). This sequence of maps should give you an idea of what
the dynamics of the R\"ossler system is in this regime. How does it
compare to that of the logistic map? 
 
\end{enumerate}

\item {\bf Universality of Unimodal Maps}

From Strogatz do problems 10.6.1 and 10.6.2 (note the introductory paragraph on p. 394).
%parametrically driven system: amplitude equation + simulation + chaos?
 
\end{enumerate}


\end{document}
 
 
