Self-Organization and Emergence in Complex Dynamical Systems

Complex computing systems begin to overwhelm the capacities of software developers and administrators. Self-organization has been a successful strategy of evolution to handle the increasing complexity of organisms with the emergence of novel structures and behavior. Thus, self-organization and emergence are fundamental concepts of organic computing. But these concepts are often used in a more or less intuitive and fuzzy manner. In the theory of complex systems and nonlinear dynamics, self-organization and emergence can be mathematically defined. Actually, these concepts are independent of biological applications, but universal features of dynamical systems. We get an interdisciplinary framework to understand self-organizing complex systems and to ask for applications in organic computing. 1 From Linear to Nonlinear Dynamics A dynamical system is a time-depending multi-component system of elements with local states determining a global state of the whole system. In a planetary system, for example, the state of a planet at a certain time is determined by its position and momentum. The states can also refer to moving molecules in a gas, the excitation of neurons in a neural network, nutrition of organisms in an ecological system, supply and demand of economic markets, the behavior of social groups in human societies, routers in the complex network of the internet, or units of a complex electronic equipment in a car. The dynamics of a system, i.e. the change of system’s states depending on time, is represented by linear or nonlinear differential equations. In the case of nonlinearity, several feedback activities take place between the elements of the system. These manybodies problems correspond to nonlinear and non-integrable equations with instabilities and sometimes chaos [Ma04].