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Bodo Werner: Research

Numerical analysis of bifurcation problems and of dynamical systems.


Bifurcation Analysis of Microscopic Traffic problems

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


Hopf bifurcation, bordering methods

One of the most prominent codimension-1 bifurcations is the Hopf bifurcation for differential equations or maps. The following paper of mine treats the detection and computation of Hopf points and the numerical path following of Hopf curves in two-parameter systems (See also the traffic papers below, from 2004 to today):

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


Bifurcation with underlying symmetry, Takens-Bogdanvov bifurcation and test functions

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.


Bodo Werner