Introduction: Synergetics and Models of Continuous and Discrete Active Media. Steady States and Basic Motions (Waves, Dissipative Solitons, etc.)

Abstract

In modern science, “synergetics” owes its development to the extraordinary research output provided by H. Haken coupled with the many workshops he has organized in the past three decades. In short it deals with the emergence of dynamical and/or evolutionary features in complex systems where the total is not the mere addition of parts. Hence an emergent property belongs to a higher level of description than that of the underlying elements. Emergent properties have genuine laws of their own as is customary in biology, ecology, sociology, etc., and indeed engineering, chemistry and physics. Already thermodynamics is a science of emergence (of cooperative properties) in underlying atoms, and statistical mechanics is the bridge between the micro and macro (or even meso) levels. Equilibrium critical phenomena and phase transitions provide paradigmatic examples of transitions and cooperativity in nonevolving systems, and they define an area of physics rather well understood both at the emergent level and in the way emergent, macroscopic and phenomenological properties originate from the, microscopic, lower level. It is one of the areas where from first principles we know the recipe to account for emergent, synergetic, properties.The reductionism from the emergent to the lower level is far from trivial and clearly indicates that, although the basics are to be found at the microlevel, understanding the emergence of cooperative properties demands a great deal of imagination and, on occasion, computer power.