Bifurcations in Particle Physics and in Crystal Growth

Abstract

In the course of structure formation, periods of slow and steady development are invariably separated by sudden and discontinuous changes in which new spatio-temporal patterns are created. The latter are triggered when, in the nonlinear domain, competing but continuously driving forces enter conflicting regimes. Such spontaneous changes can be described geometrically in terms of bifurcations. In the vicinity of a bifurcation point a system becomes extremely sensitive to small ambient factors like imperfections, external fields or fluctuations. This enhances the system’s ability to perceive its environment and to adapt to it by forming preferred patterns or modes of behavior. Many of the bifurcation processes observed in different systems are qualitatively similar and universal in the sense of being largely independent of system details. This calls for a topological description of the phenomena under consideration which defines, by the notion of equivalence, what it means for two bifurcation processes to be qualitatively similar. The same notion also allows us to account for the basic, though often forgotten fact that physical systems are structurally stable, i.e., that they preserve their quality under small perturbations. This persistence property guarantees that today’s experiment reproduces yesterday’s result. We do not know how it got that way. But, surprisingly enough, the fundamental topological invariance principle provided by this requirement of structural stability enables us to describe and to classify the bifurcation processes that underly the spontaneous formation of structure on geometrical grounds alone, irrespective of their particular physical origin.