## Bodo Werner: Research## Numerical analysis of bifurcation problems and of dynamical systems. |

**Bifurcation analysis of a class of 'car following' traffic
models** with I. Gasser, G. Sirito, Physica D: Nonlinear Phenomena,
Vol 197/3-4 pp 222-241, 2004.

**
On the influence of aggressive driving behavior on the dynamics of traffic flow,
** with T. Seidel, I. Gasser, G. Sirito,
Proceedings in Applied Mathematics and Mechanics PAMM (Wiley-Interscience Online Journal), vol. 5 (1), 693-694 (2005).

**
Car Following Models for Phenomena on the Highway,**
with T. Seidel, I. Gasser,
Proceedings 17th International
Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering (2006).

**Bifurcation Analysis of a class of Car-Following Traffic Models II:
Variable Reaction Times and Aggressive Drivers** with I. Gasser, T. Seidel, G. Sirito,
Bulletin of the Institute of Mathematics, Academia Sinica, Volume 2/2, pp 587-607 (2007).

**Microscopic Car-Following Models Revisited: From Road Works to Fundamental Diagrams**
with T. Seidel, I. Gasser, SIAM J. Appl. Dyn. Syst. 8, 1305–1323 (2009).

**Dynamical phenomena induced by bottleneck** with I. Gasser,
Phil. Trans. R. Soc. A 368, 4543-4562 (2010) (Special Issue Highway Traffic - Dynamics and Control).

**
An asymptotic numerical analysis of Hopf periodic traveling waves for a microscopic traffic problems**
Hamburger Beiträge zur Angewandten Mathematik 2011-12

**
Analysis of quasi-POMs in a microscopic traffic model**
Hamburger Beiträge zur Angewandten Mathematik 2011-15

**Computation of Hopf bifurcation with bordered matrices**,
SIAM J. Numer. Anal. 33, 435-455, 1996.

The basic numerical technique I am using is that of
**bordered systems**
which dates back to a paper of Griewank-Reddien 1986 and which is also
promoted very much by my friends Willy Govaerts (Ghent) and
Vladimir Janovsky (Prague). The theoretical question about the
possibility of choosing the bordering continuously dependent
on the singular matrices being bordered
has been addressed (and negatively answered) by

** Continuous bordering of matrices and continuous matrix
decompositions** (with W. Govaerts), Numer. Math. 70, 303-310, 1995.

My doctorate student, Haihong Xu, has adressed the problem of the numerical computation of degenerate Hopf bifurcation points, particularely those points on a Hopf curve, where the bifurcating periodic orbits lose their stability. With other words: where there is a change from super- to subcritical bifurcation. In cooperation with Vladimir Janovsky we have written the paper

** Numerical computation of degenerate Hopf bifurcation points.**
Z. Angew. Math. Mech. 78 (1998) 9, 1-15

The bordering technique has been applied to problems with symmetry in

** The numerical Analysis of Bifurcation problems with symmetries based on
bordered Jacobians**
Proceedings of the AMS--SIAM Summer Seminar 1992,
Allgower, Georg, Miranda (Eds.), Exploiting Symmetry in Applied and
Numerical Analysis, Lectures in Applied Mathematics 29,443-457,
1993.

This is one example of my interest in bifurcations with symmetry, mostly restricted to steady state and Hopf bifurcations. This interest started with pitchforks for problems with simple reflection symmetries,

**Computation of symmetry breaking bifurcation points
** (with A. Spence),
SIAM J. Numer. Anal. 21, 388-399, 1984

and has been continued with symmetries of more general groups,

** Computational methods for bifurcation problems with symmetries -
with special attention to steady state and Hopf bifurcation points**
(with
M. Dellnitz),
J. of Comp. and Appl. Math. 26, 97-123, 1989.

and

** Symmetry adapted block diagonalization in equivariant steady
state bifurcation problems ant its numerical applications**
(with P. Stork),
Advances in Mathematics (China), Vol.20,4 p. 455-487, 1991.

and

**Some remarks on period doubling in systems with symmetry**
(with N. Nicolaisen),
ZAMP 46, p. 566-579, 1995.

Certain codimension-2 bifurcations - also for problems with symmetries - have been treated in

**Computation of Hopf branches bifurcating from Takens-Bogdanov points
for problems with symmetries** (with V. Janovsky),
In: Bifurcation and Chaos
(R.Seydel, F.W.Schneider, T.Küpper, H.Troger, eds.),
ISNM 97, 377-388, Birkhäuser, 1991.

and

**Group Theoretical Mode Interactions with Different Symmetries**
(with
Karin Gatermann),
Intern. J. of Bifurcation and Chaos 4, 177-191, 1994.

The analysis of Takens--Bogdanov bifurcation points, also of higher codimension, as far as Hopf and fold points are concerned, is the aim of

**Analysis of Takens--Bogdanov--Bifurcation with
characteristic functions** (with Lijun Yang),
Nonlinear Anal., Theory Methods Appl. 26, 363-381, 1996.

The basics
of my bifurcation software were *test functions*, these are real functions
to be evaluated during the path following, which detect bifurcations
by a simple sign change. The principles of these test functions
(the name stems from Rüdiger Seydel) are investigated in

**Test Functions for Bifurcation Points and Hopf Points in
Problems with Symmetries
**,
In Allgower, Böhmer, Golubitsky (eds): Bifurcation and Symmetry, Birkhäuser,
ISNM 104, 317-327, 1992.

The project Invariant Curves, Circle Maps and their Numerical Analysis:

** Discretization of circle maps ** (with N. Nicolaisen),
Z. angew. Math. Phys. 49 (1998) 869-895

** The spectrum of the Frobenius-Perron operator
and its discretization for circle diffeomorphisms ** (with
N. Lehmkuhl and W. Hotzel),
Journal of Dynamics and Differential Equations, April 2002, vol. 14, iss. 2, pp. 443-461.