Complex systems

Interactions of individual components in a system determine an overall emergent functionality that does not independently exist. Complex systems need energy to sustain their behaviour; minute changes in one component can have consequences for the system as a whole. Theories must cover ranges of temporal or spatial scales, the interplay between the individual history and universal features, and the self-organization of elements and of collective phenomena.

Systems are composed of well- defined components that, when integrated, act together to form a functioning entity. Identifying a system or a hierarchy of systems requires a certain level of abstraction and simplification. Simple systems have few components, and their behaviour is fully understandable and predictable. Consider a ball falling through the air; this simple system consists of the ball falling under the influence of the gravitational force with a viscous drag. Add more balls and this system becomes complex: vortices from each ball affect the fall of the other balls in a manner that cannot be simply extrapolated from the fall of a single ball.

Complex systems have many components that interact to create a functioning unit without an overarching regulatory body1. It is the self-organized cooperative interaction of the individual components that determines the emergent functionalities.


Complex self-organizing systems are prevalent in nature. They are part of our everyday lives in areas ranging from ecology, sociology and economics, to biology, medicine, chemistry and physics. Many systems share similar spatio-temporal structures. For example, one might ask whether the large-scale periodic structure of sand dunes is related to the nanoscale patterns of polymer microstructures, whether the turbulence is similar in stars and in a stirred coffee cup or whether cardiac fibrillation is related to the aggregation dynamics of social amoebae. It is remarkable that these statements are true.

Numerical simulation and advanced experimental tools now allow detailed quantitative investigations of the spatio-temporal evolution of complex systems, including not only spatially extended systems2, but also highly coupled networks. This is well illustrated by the example of epidemic outbreaks and the subsequent geographic spreading of infectious disease, which incidentally mirrors the transportation of banknotes.

Higher organisms develop through the collective organization of many cells. The biological cells themselves are complex systems that use highly evolved genetic regulatory schemes to create structures that can dynamically maintain themselves or contribute to a higher multicellular organism. These regulatory schemes in turn integrate complementary types of information3. At both the single-cell and multi-cell levels there remains a fundamental challenge to understand the principles behind the developmental process that lead to the formation of complex patterns and morphologies4 (Fig. 1).


Among the organs of the human body, the functions of the heart and the brain are highly complex in that their operation emerges from the collective dynamics of millions of strongly interacting cells, which are well organized in their geometrical structure and connectivity. In heart muscle, the propagation of a nonlinear wave pulse — the cardiac action potential — controls the contraction. Usually the propagation is well organized and the heart functions as an efficient biological pump. During cardiac fibrillation, vortex-like rotating waves of electrical activity result in spatio-temporal chaotic excitation and contraction patterns5 (Fig. 2). This electro-mechanical malfunction causes an estimated 738,000 deaths per year in Europe alone. Yet the physical mechanisms underlying the dynamics and control continue to pose a fundamental scientific riddle.

Similarly, in the brain, the propagation of a nonlinear wave pulse — namely, the neural action potential — is the basis of its computational and memory power. Here the high degree of interconnectivity and topological complexity of the neuronal network results in the coordinated activity of millions of interacting nerve cells that comprise the human mind. The real-time processing of inputs interacts with the slower process of learning through activity-dependent changes of synaptic response6,7. The workings of the brain, however, cannot be fully understood if viewed only as a complex dynamical system. For example, it is possible to describe the brain’s processing of hierarchically structured sequences (such as sentences) at the neural-systems level as an initial phase of local structure building, immediately followed by a second phase of creating hierarchical dependencies8. Therefore, understanding the operation of the mind also requires describing and analysing its emergent information processing functions. This specific ability to extract rules underlying hierarchical dependencies appears to differentiate humans from non-human primates. Advances in many aspects of neural computation have been addressed by pattern formation, statistical inference and optimal decision-making, borrowing from the mathematical language of statistical physics9.


Complex-systems research also encompasses artificially created systems that are changing the environment on a scale never seen before. Can the Earth’s resources be better utilized?

Humankind has made significant progress in increasing the productivity, selectivity and sustainability of chemical and biotechnological production processes. Moving forwards, further breakthroughs in process systems engineering are needed in order to make the transition from fossil fuels and petrochemical feed stocks to renewable materials and energy. Theoretical and experimental approaches need to be developed for the analysis and design of chemical and biochemical production systems, with a focus on their inherent multilevel structure. New technological and production processes based on complex-systems research promise to close carbon dioxide cycles, enhance efficiency, and incorporate new functionality in materials and products10.


Turbulence manifests itself in myriad ways across the physical, chemical, engineering and even biological sciences. It occurs whenever fluid viscous forces are smaller than the dominant driving forces of the flow; in practice this includes most natural and technological flows11. Turbulence is a fundamental topic in fluid and dynamical systems research that raises basic questions in both applied mathematics and informatics.

The study of turbulence thus unites many types of researcher, from earth scientists studying the terrestrial biosphere and the exchange of heat, momentum and matter within the forest canopy, to physicists exploring strategies for confining plasma in experimental fusion reactors, and even mathematicians developing fundamental new numerical methods.

Advanced understanding of complex systems will open new paths for translating fundamental scientific into practical functions. It has applications in subject areas ranging from life sciences and medicine, physics, chemistry and engineering, to social, economic and cognitive sciences. The challenge for scientists is to reconcile the differences and similarities, and the shared and disparate features, across physical, biological and chemical systems that are natural and artificial.

Prediction and control of pandemics could be improved by complex-systems research at the Max Planck Institute for Dynamics and Self-Organization. Infected persons transport bacteria and viruses across great distances, transmit them to others and spread the pandemic. Researchers evaluated data from the popular banknote-tracking website ( and showed that a two-parameter random-walk model can accurately describe human travelling behaviour mathematically (Brockmann, D. et al. Nature 439, 462, 2006).

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