Introduction

Abstract

One of the most astonishing aspects in nature is the occurrence of universal behavior, i. e. behavior which occurs in many classes of physical systems. An example is represented by the phenomenon of period-doubling, an intermediate process which very often connects highly regular periodic motions with chaotic motions in oscillatory systems like dropping watertaps, technical oscillators or the human heart. For the human heart this means that a beating heart changes its state in such a way that first an alternating series of strong and weak heartbeats and then a totally irregular motion occurs. Such dramatic changes of states are normally called phase transitions. Phase transitions of several kinds occur when specific control parameters cross critical values. Such phase transitions are observable in the thermodynamic equilibrium as well as far from thermodynamic equilibrium. Examples of equilibrium phase transitions are phase transitions of matter like the transition of water from the liquid into the gas state. Examples of non-equilibrium phase transitions are the laser transitions. For example, special laser transitions separate non-lasing from lasing states. Other non-equilibrium phase transitions are biological transitions like the above described heart transition or the change of the movement state of walking animals like the transition of horses’ gait from trot to gallop. A widely used classification scheme for phase transitions is the scheme of Ehrenfest, i.e. the separation of phase transitions into phase transitions of first and second order. First order transitions are transitions of matter, i. e. melting, boiling or condensing.