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\centerline{\large \bf Interdisciplinary Nonlinear Dynamics (438)}
\centerline{Fall 2001}
\centerline{Hermann Riecke}
\vspace{.5cm}
 
\centerline{Problem Set 6}

\vspace{.5cm}

\centerline{For Discussion Section November 9}
 
\vspace{.5cm}
  
\begin{enumerate}
 
\item Do problem 3c and 3d from previous homework assignment.

\item {\bf An Interesting Dynamical System\footnote{This is one of the interesting
situations that can arise at a codimension-2 bifurcation, i.e. 
when two modes go unstable at the same parameter value
 (cf. Guckenheimer \& Holmes, {\it Nonlinear Oscillations,
Dynamical Systems and Bifurcations of Vector Fields}, Ch. 7.3)}}

Consider the dynamical system 
\bea
\frac{d x}{dt} &=& y,\label{e:doublex}\\
\frac{d y}{dt} &=& \alpha x + \beta y + a x^3 - x^2 y.\label{e:doubley}
\eea
In this problem you will study this system analytically and numerically. For
the numerical simulations you can use {\tt pplane5}\footnote{You may want to
set the  solver to {\tt ODE23} and set the calculation window to 10 times the
display window.}  or write your own code. {\tt pplane} has the advantage to 
provide you right away with a number of graphical options. 

\begin{enumerate}

\item Determine analytically 
the stability of the fixed point $(x,y)=(0,0)$ as a function
of $\alpha$ and $\beta$. Based on this information, 
what type of behaviors may be expected as $\alpha$ and/or $\beta$ is varied?

\item Consider the case $a=-1$. 

\begin{enumerate}
\item Solve (\ref{e:doublex},\ref{e:doubley}) numerically.
 What kind of nonlinear behavior do you find if you vary $\alpha$ across the instability 
identified above for $\beta=-1$?\\
For $\alpha=0$ and $\beta=-1$ 
identify the center manifold of the attractor numerically by starting from a 
number of different initial conditions. 
Increase now $\alpha$ to $0.05$, $0.1$, and $1$. 
What happens to the center manifold?

\item For small $\alpha$ perform analytically a center-manifold reduction of 
(\ref{e:doublex},\ref{e:doubley}), i.e. determine a center manifold 
that
describes the dynamics of this system for small amplitudes $x$, $y$, and for
small values of the bifurcation parameter $\alpha$. Go to cubic
order. Obtain the evolution equation for 
the dynamics of the system on that manifold. Compare the analytically obtained
center manifold and the fixed points on it quantitatively with your numerical results. 

\end{enumerate}

\end{enumerate}
\end{enumerate}
\end{document}
\item Consider now the case $a=+1$. 

\begin{enumerate}
\item Choose $\alpha=-1$ and increase $\beta$ across 
the instability identified above. What happens in the numerical simulation? \\
Measure the amplitude ${\cal A}$ and the frequency $\omega$ 
of the oscillations as a function of $\beta$,
increasing $\beta$ up to $\beta=0.2$. Is there a range in $\beta$ over 
which the dependence of ${\cal A}$ and $\omega$ on $\beta$ agrees with your expectation 
for such a bifurcation? \\
What happens for $\beta >0.2$?

\end{enumerate}


\item Go back to the case $a=-1$.

\begin{enumerate}

\item Study numericaly how the phase portrait changes for $\alpha=0.1$ when $\beta$ is
varied from $\beta=-1$ to $\beta=0.2$. What happens to the center manifold? 

\end{enumerate}

\end{enumerate}
\end{enumerate}
\end{document}
 
 
