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\centerline{\large \bf Interdisciplinary Nonlinear Dynamics (438)}
\centerline{Fall 2001}
\centerline{Hermann Riecke}
\vspace{.5cm}
 
\centerline{Problem Set 7}

\vspace{.5cm}

\centerline{For Discussion Section November 30}
 
 
\vspace{.5cm}
  
\begin{enumerate}
\item {\bf An Interesting Dynamical System II}

Consider again the 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
from the previous homework set.

\begin{enumerate}
\item Consider now the case $a=+1$. \\
  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 $\beta >0.2$?

\item Go back to the case $a=-1$.\\
  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}
\item {\bf Pitch-Fork Bifurcation by Multiple Times}

Consider again (\ref{e:doublex},\ref{e:doubley}). 
Use now multi-timing to analyze the steady bifurcation, which
you analyzed in Problem 1.b.ii of the last problem set using center-manifold
reduction. Do you get the same bifurcation equation?

\item {\bf Swift-Hohenberg Equation without Up-Down Symmetry}

Consider the modified Swift-Hohenberg equation
\bea
\partial_t \psi = \mu \psi -(\partial_x^2+1)^2\psi +\alpha \psi^2 -\psi^3, \qquad -\infty < x < \infty,
\qquad 
\label{e:msh}
\eea
which does not possess the `up-down' symmetry $\psi \rightarrow -\psi$.
Derive the Ginzburg-Landau equation for (\ref{e:msh}) by expanding around
the periodic solution with critical wavenumber $q_c=1$ and allowing the
amplitude to be a function of slow time {\it and} space. Show that one obtains 
a pitch-fork bifurcation despite the broken symmetry. Check that the spatial
translation symmetry of (\ref{e:msh}), i.e. invariance under $x \rightarrow
\Delta x$ for any $\Delta x$, corresponds (implies) the symmetry of the
Ginzburg-Landau equation under phase shifts $A \rightarrow Ae^{i\Delta
\phi}$. This illustrates that the origin of the pitch-fork bifurcation is the
translation symmetry of the system.


\item {\bf Numerical Solution of Ginzburg-Landau Equation}

\begin{enumerate}
\item Modify the code provided on the class web site, 
which solves the SH-equation, to solve the Ginzburg-Landau
equation
\bea
\partial_t A =(1+ic_1) \partial_x^2 A+\mu A - (1-ic_3)|A|^2A,
\eea
where $A$ is the complex amplitude of the pattern. The code uses backward Euler\footnote{Setting 
$\alpha = 0.5$ in the code turns it into a Crank-Nicholson scheme.} for the spatial derivatives, 
whereas the terms without derivatives are updated with a forward Euler step. The backward Euler method
provides unconditional stability with respect to the derivative terms. 
 Thus, you want to modify the code to solve
\bea
A_j^{n+1}=A_j^n + (1+ic_1)\frac{\Delta t}{\Delta x^2} \left(A_{j+1}^{n+1}-2A_j^{n+1}+A_{j-1}^{n+1}\right)
+\Delta t \left(\mu A_j^n -(1+ic_3)|A_j^n|^2A_j^n\right).
\eea
Note that matlab actually deals with complex variables in a straightforward
manner.\\
Plot the real part of $A$, the imaginary part of $A$, and its absolute value.

 
\item Use your code to study numerically the Eckhaus stability limits obtained
in class  from the phase equation. Do that for the case $c_1=0=c_3$, i.e. for
steady patterns. Start with a {\it periodic} pattern in a sufficiently large system
(at least 10 to 15 wavelengths) in the stable regime and let it converge to a
steady solution. Then decrease $\mu$ somewat below the expected stability
limit and perturb this non-zero solution by adding some noise. How does this
perturbed solution evolve in time and what is the final outcome of the
simulation?  

\item Just for the fun of it run the code now for $c_1=0$ and $c_3=1.4$ with some random initial condition.
In this regime the complex Ginzburg-Landau equation shows interesting behavior (cf. M. van Hecke, Phys. Rev. Lett. 80 (1998) 1896, go to {\tt prl.aps.org} to look at the paper on the web. More interesting regimes
can be found in H. Chat\'e, Nonlinearity 7 (1994) 185 ({\tt
http://www.iop.org/Journals/no})).
\end{enumerate}

%parametrically driven system: amplitude equation + simulation + chaos?

%center manifold by multi-scaling?

\end{enumerate}

\end{document}
 
 
