American Institute of Mathematical Sciences

Advanced
November  2003, 3(4): 519-540. doi: 10.3934/dcdsb.2003.3.519

Regular and random patterns in complex bifurcation diagrams

E. Kapsza 1, , Gy. Károlyi 2, , S. Kovács 3, and  G. Domokos 4,

1. 

Department of Mechanics, Matherials and Structures, Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1521 Budapest, Hungary
2. 

Center for Applied Mathematics and Computational Physics and Department of Structural Mechanics, Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1521 Budapest, Hungary
3. 

Department of Mechanics, Materials and Structures, Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1521 Budapest, Hungary
4. 

Center for Applied Mathematics and Computational Physics and Department of Mechanics, Materials and Structures, Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1521 Budapest, Hungary

Received  November 2002 Revised  May 2003 Published  August 2003

Based on a well-known discrete bifurcation problem (the discretized Euler buckling problem) displaying a highly complex bifurcation diagram, we show how to find fast, global access to the distribution patterns of classical branch-invariants (symmetry groups, nodal properties, stability characteristics), without actually computing the complex diagram. At the core of our method is a symbolic dynamics based labeling system, which can be viewed itself as a (non-classical) global invariant and from which all the classical invariants can be derived. Based on results from the theory of Brownian bridges an approximate sequence of these integer-labels can be obtained very fast, in fair agreement with the measured quantities. Similar labeling systems have been used in other problems, so we argue that our method will be useful for a wider range of boundary value problems displaying spatial complexity characterized by a mixture of regular and random patterns.
Keywords: standard map, Brownian bridge., elastic linkage, Spatial chaos.
Mathematics Subject Classification: 37M2.
Citation: E. Kapsza, Gy. Károlyi, S. Kovács, G. Domokos. Regular and random patterns in complex bifurcation diagrams. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 519-540. doi: 10.3934/dcdsb.2003.3.519
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