Regular and random patterns in complex bifurcation diagrams

Abstract

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.