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{\fontsize{9}{10}
\bf 8th. World Congress on Computational Mechanics (WCCM8)\\
5th European Congress on Computational Methods in Applied Sciences and Engineeering (ECCOMAS 2008)\\
June 30 --July 5, 2008\\Venice, Italy\\}
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{\fontsize{14}{20}\textbf{A NEW DEVELOPMENT PLATFORM FOR {PARAMETER
CONTINUATION AND }BIFURCATION ANALYSIS IN NONLINEAR DYNAMICAL SYSTEMS}}
\medskip
\textbf{*Harry Dankowicz$^{1}$ and Frank Schilder$^{2}$}
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\centerline{
\hbox{
\begin{minipage}{8.3cm}
$^1$Department of Mechanical Science and Engineering\\
University of Illinois at Urbana-Champaign\\
Urbana, IL 61801, USA \\
danko@uiuc.edu
\end{minipage}
\quad
\begin{minipage}{6.1cm}
$^2$Department of Mathematics\\
University of Surrey\\
Guildford, GU2 7XH, United Kingdom\\
f.schilder@surrey.ac.uk
\end{minipage}
}}
\vspace{0.5cm} \textbf{Key Words:} \textit{Bifurcation analysis, Parameter
continuation, Software development, Coco.}
\begin{center}
\textbf{ABSTRACT}\\[1mm]
\end{center}
A combination of theoretical and computational tools for bifurcation
analysis of dynamical systems offers distinct advantages to brute-force
forward-time simulation [9]. Such a combination enables prediction of
behavior and response without the need for a vast collection of simulations
based at distinct initial conditions. More importantly, it may offer an
understanding of, and an underlying explanation for, changes in behavior and
response that are not available through simple simulation.
A comprehensive bifurcation analysis of a dynamical system seeks to
establish the existence of characteristic classes of responses, such as
equilibria or periodic responses. In each case, this involves locating and
tracking families of such responses under variations in system parameters in
a process known as \emph{continuation} [1,5,8,10]. The study of the
robustness of particular system responses further emphasizes parameter
values where such families merge or terminate or where the stability
characteristics of the corresponding responses change.
A number of computational tools are available for bifurcation analysis of
characteristic classes of response, such as equilibria, periodic
trajectories, homo- or heteroclinic trajectories between equilibria and/or
periodic trajectories, quasiperiodic trajectories on invariant tori, and
trajectories on associated stable and unstable manifolds. These include
general algebraic and two-point boundary-value solvers for ordinary
differential equations, such as \textsc{auto} (and specialized drivers, such
as \textsc{homcont} [2], \textsc{slidecont} [3], and \textsc{tc-hat} [13]),
\textsc{matcont} [4], and \textsc{sympercon} [14]; boundary-value solvers
for delay differential equations, such as \textsc{dde-biftool} [6] and
\textsc{pdde-cont} [12]; tools for large-scale systems, such as \textsc{loca}
[11]; and implementations in \textsc{matlab} [7].
This presentation outlines a recent software development effort in \textsc{%
matlab} of a set of core toolboxes for parameter continuation of algebraic
and multi-point boundary-value problems referred to collectively as \textsc{%
coco}. In contrast with the packages referenced above, \textsc{coco} has
been formulated with emphasis on full transparency, modularity, and great
generality giving developers of bifurcation packages that rely on this
platform great flexibility and full access to the entirety of continuation
data. In addition, \textsc{coco} makes extensive use of vectorization and
bandwidth reduction for sparse-matrix operations to guarantee an optimal
execution time at or close to what can be expected of interpreted code.
Finally, \textsc{coco} includes a unique implementation of user-defined
functions that allows the continuation of solutions with nontrivial
properties, for example, periodic orbits with a given stability margin. The
presentation discusses the philosophy behind the software development, the
strategy employed in its implementation, and a number of model examples,
including continuation of periodic orbits in hybrid dynamical systems.
This material is based upon work supported by the National Science
Foundation under Grant nos. 0237370 (HD) and 0635469 (HD) and by the
Engineering and Physical Sciences Research Council under Grant no.
EP/D063906/1 (FS).
\begin{center}
\textbf{REFERENCES}
\end{center}
\begin{tabular}{lp{145mm}}
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continuation methods: An introduction}, Springer-Verlag, 1990. \\[1mm]
\hspace{-2mm}{[2]} & A.R.\ Champneys, Y.A.\ Kuznetsov, and B.\ Sandstede,
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[1mm]
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[1mm]
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SIAM J. Sci. Comput.}, Vol. \textbf{27(1)}, 231--252, 2005. \\[1mm]
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\\[1mm]
\hspace{-2mm}{[9]} & J.\ Guckenheimer, ``Computer simulation and beyond--for
the 21st century,'' \emph{Notices Amer. Math. Soc.}, Vol. \textbf{45(9)},
1120--1123, 1998. \\[1mm]
\hspace{-2mm}{[10]} & H.B.\ Keller, \emph{Lectures on Numerical Methods in
Bifurcation Problems}, Springer-Verlag, New York, 1987. \\[1mm]
\hspace{-2mm}{[11]} & A.G.\ Salinger, E.A.\ Burroughs, R.P.\ Pawlowski,
E.T.\ Phipps, and L.A.\ Romero, ``Bifurcation tracking algorithms and
software for large scale applications,'' \emph{J. Bifur. Chaos Appl. Sci.
Engrg.}, Vol. \textbf{15(3)}, 1015--1032, 2005. \\[1mm]
\hspace{-2mm}{[12]} & R.\ Szalai, G.\ Stepan, and S.J.\ Hogan,
``Continuation of bifurcations in periodic delay-differential equations
using characteristic matrices,'' \emph{SIAM J. Sci. Comput}, Vol. \textbf{%
28(4)}, 1301--1317, 2006. \\[1mm]
\hspace{-2mm}{[13]} & P.\ Thota and H.\ Dankowicz, ``On a Boundary-Value
Formulation for the Continuation of Solution Trajectories in Hybrid
Dynamical Systems and its Implementation in the Software Toolbox TC-HAT ($%
\widehat{TC}$),'' \emph{SIAM J. App. Dyn. Syst.}, in review, 2007. \\[1mm]
\hspace{-2mm}{[14]} & C.\ Wulff and A.\ Schebesch, ``Numerical continuation
of symmetric periodic orbits,'' \emph{SIAM J. Appl. Dyn. Syst.}, Vol.
\textbf{5(3)}, 435--475, 2006.%
\end{tabular}
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