FROM SYMMETRY TO COMPLEXITY: ON INSTABILITIES AND THE UNITY IN DIVERSITY IN NONLINEAR SCIENCE

Abstract

Throughout this paper I will argue that dynamical and structural instabilities are the sources of complexity and pattern formation. The argument can be accomplished by defending two theses. First, a demarcation thesis: two different approaches are predominant in mathematical sciences today — the symmetry-approach and the complexity-approach. Second, a synthesis thesis: although the two approaches differ, they can be connected and, further, to some degree integrated: instabilities are core concepts connecting the two approaches. However, in a specific sense we can say: evolution leads from symmetry to complexity by transitions across borders of instabilities. This paper will provide further arguments in favor of a structural unity in phenomenological diversity [Mainzer, 2005].