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Much of our knowledge concerning the dynamics of specific nonlinear
systems comes from numerical simulations. I am interested in developing
techniques - this includes both theory and algorithms - that allow us
to efficiently use the computer to rigorously verify the observed
dynamics.
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Low Dimensional Dynamics
- Z. Arai
W. Kalies, H. Kokubu,
K. Mischaikow, H. Oka, and Pl. Pilarczyk, A Database Schema for the
Analysis of Global Dynamics of Multiparameter Systems, SIAM J. Applied Dyn. Syst., 8 757 (2009).
- Z. Arai
and
K. Mischaikow, Rigorous
Computations
of Homoclinic Tangencies
(The final version can be found in SIAM
Dynamical Systems, 5
(2006) 280-292.).
- M.
Gameiro
, T.
Gedeon
, W. Kalies, H. Kokubu, K.
Mischaikow, and H. Oka, Topological
Horseshoes
of Travelling Waves for a Fast-Slow Predator-Prey System
(The final version can be found in Jour.
Dyn. Diff. Eqns., 19
(2007) 623-654.).
- S. Day,
O.
Junge, and K.
Mischaikow, Towards
Automated Chaos Verification, (To appear in Proc. Equadiff 2003)
- K. Mischaikow, M.
Mrozek , and A.
Szymczak , Chaos in
the Lorenz equations: A computer assisted proof. Part III: Classical
Case Parameter Values
- K. Mischaikow and M.
Mrozek , Chaos
in the Lorenz equations: A computer assisted proof. Part II: Details (The
final
version can be found in Mathematics of Computation, 67
(1998) 1023-1046)
-
Infinite Dimensional Dynamics
- S. Maier-Paape, K. Mischaikow, and T. Wanner, Structure of the
Cahn-Hilliard Equation on the square, Int.
J. Chaos Bif., 17
(2007) 121-1263.
- S. Day,
J.-P. Lessard, and K.
Mischaikow, Validated
continuation
for equilibria of PDEs,
(The final version can be found in SIAM
Numerical Anal., 45,
2007, 1398-1424).
- S. Day,
Y.
Hiraoka, K.
Mischaikow, and T. Ogawa, Rigorous
Numerics
for Global Dynamics: a study of the Swift-Hohenberg
equation (The final version can be found in SIAM
Dynamical Systems, 4,
2005).
- S. Day,
O.
Junge, and K.
Mischaikow, A
Rigorous
Numerical Method for the Global Analysis of Infinite
Dimensional Discrete Dynamical Systems The final version can
be found in SIAM
Dynamical Systems, 3,
2004).
- P.
Zgliczynski
and K. Mischaikow, Rigorous
Numerics
for Partial Differential Equations: the Kuramoto-Sivashinsky
equation (The final version can be found in Foundations
of
Comp. Math. 1 255-288.)
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