The theory of nonlinear complex systems has become a proven problem-solving approach in the natural sciences from cosmic and quantum systems to cellular organisms and the brain. Even in modern engineering science self-organizing systems are developed to manage complex networks and processes. It is now recognized that many of our ecological, social, economic, and political problems are also of a global, complex, and nonlinear nature. Modern evolutionary economics can be modelled in the framework of complex systems and nonlinear dynamics. Historically, evolutionary economics was inspired by Schumpeterian concepts of business cycles and innovation dynamics. What are the laws of sociodynamics? What can we learn from nonlinear dynamics for complexity management in social, economic, financial and political systems? Is self-organization an acceptable strategy to handle the complexity in firms, institutions and organizations? The world-wide crisis of financial markets and economies is a challenge for complexity research. Misleading concepts of linear thinking and mild randomness (e.g. Gaussian distributions of Brownian motion) must be overcome by new approaches of nonlinear mathematics (e.g. non-Gaussian distribution), modelling the wild randomness of turbulence at the stock markets. Systemic crises need systemic answers. Nevertheless, human cognitive capabilities are often overwhelmed by the complexity of nonlinear systems they are forced to manage. Traditional mathematical decision theory assumed perfect rationality of economic agents (homo oeconomicus). Herbert Simon, Nobel Prize laureate of economics and one of the leading pioneers of systems science and cognitive science, introduced the principle of bounded rationality. Therefore, we need new insights into the factual microeconomic behaviour of economic agents by methods of humanities, cognitive and social sciences, which are sometimes called ‘experimental economics’. Social and economic dynamics are interdisciplinary challenges of modern complexity research.

An approach to Complexity from the perspective of fundamental science is outlined, drawing on the cross-fertilization of concepts and tools from nonlinear dynamics, statistical physics, probability and information theories, data analysis and numerical simulation. Emphasis is placed on the intertwining between different levels of description and on the probabilistic dimension of complex systems, in connection with the issue of prediction.

We will illustrate the universality of strongly coupled plasmas by discussing two new forms of these plasmas, which have only recently become possible to create and observe in the laboratory. They exhibit a wealth of intriguing complex behavior, which can be studied experimentally, in many cases for the first time. Plasmas, gases of charged particles, are universal in the sense that certain properties of complex behavior only depend on ratios of characteristic parameters of the plasma, not on the parameters themselves. Therefore, it is of fundamental and far-reaching consequence, to be able to create and observe a strongly coupled plasma since its behavior is paradigmatic for an entire class of plasmas.

Chemistry has developed from molecular chemistry, mastering the combination and recombination of atoms into increasingly complex molecules, to supramolecular chemistry, harnessing intermolecular forces for the generation of informed supramolecular systems and processes through the implementation of molecular information carried by electromagnetic interactions. Supramolecular chemistry is actively exploring systems undergoing self-organization, i.e. systems capable of spontaneously generating well-defined functional supramolecular architectures by self-assembly from their components, on the basis of the molecular information stored in the covalent framework of the components and read out at the supramolecular level through specific molecular recognition interactional algorithms, thus behaving as programmed chemical systems. Supramolecular entities as well as molecules containing reversible bonds are able to undergo a continuous change in constitution by reorganization and exchange of building blocks. This capability defines a Constitutional Dynamic Chemistry (CDC) on both the molecular and supramolecular levels. CDC introduces a paradigm shift with respect to constitutionally static chemistry. It takes advantage of dynamic constitutional diversity to allow variation and selection and thus adaptation. The merging of the features of supramolecular systems – information and programmability; dynamics and reversibility; constitution and structural diversity – points towards the emergence of adaptive chemistry. A further development will concern the inclusion of the arrow of time, i.e. of non-equilibrium, irreversible processes and the exploration of the frontiers of chemical evolution towards the establishment of evolutive chemistry, where the features acquired by adaptation are conserved and transmitted. In combination with the corresponding fields of physics and biology, chemistry thus plays a major role in the progressive elaboration of a science of informed, organized, evolutive matter, a science of complex matter.

Our intuition assumes that there is a centre in our brain in which all relevant information converges and where all decisions are reached. To neurobiologists, the brain presents itself as a highly distributed system in which a very large number of processes occur simultaneously and in parallel without requiring coordination by a central convergence centre. The specific architecture resembles, in many respects, small world networks and raises the question of how the multiple operations occurring in parallel are bound together in order to give rise to coherent perception and action. Based on data obtained with massive parallel recordings, the hypothesis will be forwarded that temporal coherence serves as an important organizing principle and that this coherence is achieved by the synchronization of oscillatory activity in distinct frequency bands.

Human life changes with time. It seems therefore obvious that most of the phenomena that psychology and psychotherapy are concerned with are dynamic in nature. For human development processes, human change and learning processes, the dynamics and prognosis of mental disorders, problems manifesting in social systems such as couples, families, teams, or the question of how psychotherapy works, self-organization is ubiquitous. In the context of self-organization, complexity is a quality of changing patterns and patterns of change, produced by nonlinear coupled systems.

Over the last decade, we have witnessed the birth of a new movement of interest and research in the study of complex networks. These networks often have irregular structural properties, but also encompass rich dynamics. The interplay between the network topological structure and the associated dynamics attracts a lot of interest. In this research line, we propose a network approach to dealing with complex dynamics, in particular with synchronization dynamics. From the methodological perspective, this approach requires novel ideas from nonlinear sciences, statistical physics and mathematical statistics. Furthermore, we show applications in different disciplines, from earth sciences to brain dynamics. The complex network’s approach is an interdisciplinary topic and could be promising for the understanding of complexity from a systems level.

Climate exhibits a vast range of dissipative structures. Some have characteristic times of a few days; others evolve over thousands of years. All these structures are interdependent; in other words, they communicate. It is often considered that the only way to cope with climate complexity is to integrate the equations of atmospheric and oceanic motion with the finest possible mesh. Is this the sole strategy? Aren’t we missing another characteristic of the climate system: its ability to destroy and generate information at the macroscopic scale? Paleoclimatologists consider that much of this information is present in palaeoclimate archives. It is therefore natural to build climate models such as to get the most of these archives. The strategy proposed here is based on Bayesian statistics and low-order non-linear dynamical systems, in a modelling approach that explicitly includes the effects of uncertainties. Its practical interest is illustrated through the problem of the timing of the next great glaciation. Is glacial inception overdue or do we need to wait for another 50,000 years before ice caps grow again? Our results indicate a glaciation inception in 50,000 years.

Complex dynamic behaviour in terms of chaotic motion, catastrophic events or other seemingly irregular and unexpected features of and in theoretical economic models – aimed at describing real-world phenomena – are nowadays known as a common property of many nonlinear approaches to an understanding of the motion of actual time series, such as inflation rates, unemployment figures, and many other – mainly macroeconomic – economic variables. Since most existing models in economic dynamics are constructed in the tradition of classical mechanics, this result does not appear as a real surprise. However, the real ‘complexity challenge’ for economic theory still persists in identifying the complex structure of economic reality, which cannot be satisfactorily represented by simple deterministic laws of motion, although such ‘laws’ might possess the possibility of very complicated dynamic motion.

This contribution summarizes some typical features of complex systems such as non-linear interactions, chaotic dynamics, the ``butterfly effect’’, phase transitions, self-organized criticality, cascading effects, and power laws. These imply sometimes quite unexpected, counter-intuitive, or even paradoxical behaviors of socio-economic systems. A typical example is the faster-is-slower effect. Due to their tendency of self-organization, complex systems are often hard to control. Instead of trying to control their behavior, it would often be better to pursue the approach of guided self-organization, i.e. to use the driving forces of the system rather than to fight against them. This is illustrated by the example of hierarchical systems, which need to fulfill certain principles in order to be efficient and robust in an ever-changing environment. We also discuss the important role of fluctuations and heterogeneity for the adaptability, flexibility and robustness of complex systems. The presentation is enriched by a number of examples ranging from decision behavior up to production systems and disaster spreading.

For the architecture theorist Charles Jencks, Frank Gehry’s Guggenheim Museum in Bilbao, Peter Eisenman’s Aronoff Center in Cincinnati, and Daniel Libeskind’s Jewish Museum in Berlin are architectural replies to the question of the cultural outgrowths of ‘complexity science’. In the light of new technologies being used in architecture, it seems necessary to explore Jencks’s position from new perspectives and to ask: in the context of architectural production, is it possible to discuss complexity not only as an artistic-aesthetic category, but also as a fundamental technical-constructive idea? Contemporary information technologies confront architectural-theoretical discourses with developments that call for an expanded theoretical instrumentarium. It remains unclear which architectural language might be used best to approach the concept of complexity associated with information technologies.