# Complex systems

**Complexity** is emerging as a post-Newtonian paradigm for approaching from a unifying point of view a large body of phenomena occurring in systems constituted by several subunits, at the crossroads of physical, engineering, environmental, life and human sciences. For a long time the idea prevailed that the perception of systems of this kind as complex arises from incomplete information, in connection with the presence of a large number of variables and parameters masking the underlying regularities. Over the years experimental data and theoretical breakthroughs challenging this view have become available, showing that *Complexity* is on the contrary rooted into the fundamental laws of physics. This realization opens the way to a systematic study of Complexity, which constitutes today a highly interdisciplinary, fast growing branch of science, drawing on the cross-fertilization of concepts and tools from nonlinear dynamics, statistical physics, probability and information theories, data analysis and numerical simulation.

Traditionally, fundamental science explores the very small and the very large, both of which lie beyond man's everyday perception. The uniqueness of complex systems is that they have to do with a class of phenomena of fundamental importance in which the system and the observer may evolve on comparable time and space scales.

## Phenomenology of Complexity

A system perceived as complex induces a characteristic phenomenology the principal signature of which is the multiplicity of possible outcomes, endowing it with the capacity to choose, to explore and to adapt. This process can be manifested in different ways.

- The emergence of traits encompassing the system as whole, that can in no way be reduced to the properties of the constituent parts. Emergent properties reflect the primordial role of interactions between parts. They are manifested by the creation of self-organized states of a hierarchical and modular type, where order and coherence are ensured by a bottom-up mechanism rather than through a top-down design and control. Classical laboratory scale examples of this behavior are found in fluids under stress (e.g. Rayleigh-Benard cells in a fluid heated from below) and in open chemically reacting systems (e.g. bistability, oscillations, Turing patterns and wave fronts in the Belousov-Zhabotinski reaction and related systems). Further examples, in naturally occurring systems, are provided by the communication and control networks in living matter, from the genetic to the organismic to the population level.
- The intertwining, within the same phenomenon, of large scale regularities and seemingly erratic evolutionary trends. This coexistence of order and disorder raises the issue of predictability of the future evolution of the system at hand on the basis of the record available. Typical examples are provided by the atmosphere in connection with the familiar difficulty to issue reliable weather forecasts beyond a horizon of a few days, as well as by extreme geological and environmental phenomena such as earthquakes and floods. Human systems such as traders in stock markets influencing both each other and the market itself are also confronted to unexpected crises and collapses, despite the rationality supposed to prevail at the individual level. Fractals, deterministic chaos and its extreme form of fully developed turbulence provide valuable prototypes of coexistence of order and disorder in time and space.

In addition to its macroscopic level manifestations complexity is also ubiquitous at the microscopic level. Systems with built-in disorder like glassy materials give rise to a rich variety of evolutionary processes driven by microscopic level interactions. A variety of systems operating on the nanometer scale exhibit complex behaviors like energy transduction and anomalous transport, arising from the interplay between microscopically generated spontaneous fluctuations and systematic environmental constraints. The very origin of irreversibility is related to the intrinsic complexity of the dynamics of the atoms constituting a macroscopic system under the effect of their mutual interactions.

## Foundations of Complexity research

Emergent properties reflect the existence of different levels of description, obeying to their own laws, expressed in terms of appropriately defined variables. To understand the origin of these laws starting with the fundamental laws governing the elementary constituents of matter it is necessary to invoke the concepts of nonlinearity and of constraint.

Nonlinear systems subjected to constraints associated, for instance, with the distance from the state of thermodynamic equilibrium can be analyzed in depth by the methods of nonlinear dynamics, a special branch of physical and mathematical science. As the constraint removes gradually the system from a reference state (like e.g. the state of thermodynamic equilibrium, known to be unique and to lack any form of large scale structure and activity far from phase transition points), several qualitatively different regimes are generated. To these regimes correspond well-defined mathematical objects, the attractors, each of which is reached irreversibly from a set of initial states that is specific to it, referred to as basin of attraction. The evolutionary landscape is thus partitioned in cells in which different destinies are realized, and yet, this is not in any sort of contradiction with the deterministic character of the underlying evolution laws. The nature, number and accessibility of the attractors can be modified by varying the constraints. These variations are marked by critical situations where the evolutionary landscape changes in a qualitative manner as new kinds of behavior are suddenly born beyond a threshold value of the constraint, notably through the mechanisms of instability and of bifurcation. Criticalities and bifurcations confer to the system the possibility to choose, to adapt and to keep the memory of past events, since different pathways can be followed under identical ambient conditions (see Figure 1). They are best captured by switching to new, collective variables referred to as *order parameters* which in many instances turn out to obey to universal laws known as normal forms. As a rule, however, there is no universal, exhaustive classification of all the evolution scenarios: the evolution of complex systems is an open ended process that remains nevertheless compatible with the causal and deterministic character of the laws of nature. It should be stressed that a system close to a criticality displays an enhanced sensitivity, since minute differences in the value of the constraint and in the choice of the initial state will generate evolutions towards different regimes. In contrast, any given regime/attractor is robust towards perturbations.

Among the regimes that can be realized by nonlinear systems the regime of deterministic chaos is of special interest, as robustness and sensitivity are here in permanent coexistence: while the attractor descriptive of chaos is reestablished once perturbed, initially nearby states on the attractor diverge subsequently in an exponential fashion. This sensitivity to the initial conditions highlights further the issue of predictability of certain phenomena associated with complex systems, even if these are governed by deterministic laws. It also provides yet another generic mechanism of evolution in which the future remains largely open.

On the basis of the foregoing it appears that the multiplicity, sensitivity and intrinsic randomness of complex systems cannot be fully accounted for by the traditional deterministic description, in which one focuses on the detailed pointwise evolution of individual trajectories. The probabilistic approach offers a natural alternative. A fundamental point is that the evolution of systems composed of several subunits and undergoing complex dynamics can be mapped into a probabilistic description in a self-consistent manner, free of heuristic approximations. The evolution of the relevant variables takes then a form where the values featured in the deterministic description are modulated by the random fluctuations generated by the strongly unstable dynamics prevailing at the microscopic level. This accentuates further the variety of the behaviors available and entails that the probability distribution functions, rather than the variables themselves, become now the principal quantities of interest. They obey to evolution equations like the master equation or the Fokker-Planck equation which are linear and guarantee (under mild conditions) uniqueness and stability of the stationary probability densities, contrary to the deterministic description which is nonlinear and generates multiplicity and instability.

## Characterization of complex systems

The conjunction of the probabilistic and deterministic descriptions as well as of the macroscopic and microscopic views opens the way to a multilevel approach at the heart of present day complexity research as summarized below.

### Predicting complex systems

Thanks to their inherent linearity and stability, probability distributions can be used to make reliable predictions on the future occurrence of events conditioned by the states prevailing at a certain time. Of special interest is the prediction of extreme events, of the recurrence of states of a certain type and of the crossing of thresholds. The states concerned by probabilistic predictions are to be understood in a coarse grained sense of ensembles of nearby pointwise states (as those featured in the deterministic approach). This approach finds nowadays intensive use in operational weather forecastings, where it is known under the name of ensemble forecasting.

### Complexity, entropy and generalized dimensions

A probabilistic process can be characterized by a hierarchy of entropy-like quantities, describing the amount of data needed to identify a particular state of the system (Shannon entropy) or a sequence thereof (block or dynamical entropies) with a prescribed resolution. The Kolmogorov-Sinai entropy is the infinite resolution limit of block entropies and characterizes the degree of dynamical randomness of the system. Entropy-like quantities generate also a hierarchy of dimension-like quantities, generally fractal, providing a useful geometric characterization of complexity.

### Complexity and information

The probabilistic description of complex systems offers a representation in terms of sequences of states that can be regarded as symbols, or letters of an alphabet. In this view, complex systems are regarded as sources and processors of information. Symbolic sequences can be characterized by the length of the minimal algorithm that allows the observer to reconstitute them, referred to as algorithmic complexity or Kolmogorov-Chaitin complexity. Fully random sequences are the most complex ones in this perspective but it is believed that natural complexity lies between full order and full randomness adding, in a sense, a dynamic, "nonequilibrium" dimension to the concept of algorithmic complexity.

### Scales, correlations, self-similarity

An alternative characterization of complex systems is in terms of correlations, a set of quantities describing, in an averaged way, how a system keeps in time and space the memory of a perturbation inflicted initially on one of its parts. As a rule the onset of complex behaviors is marked by the generation of long range correlations which in some extreme situations are scale free in the sense of displaying no privileged characteristic time or space scale. The associated probability distributions display then, in turn, power law tails. These features are referred to as self-similar, or fractal laws.

### Simulating complex systems

Direct simulation of a process of interest, rather than the integration of a set of underlying evolution equations, is an indispensable element in the study of complex systems. Starting from a minimal amount of initial information deemed to be essential, different scenarios compatible with this information are explored. Generic aspects of complex behaviors observed across a wide spectrum of fields (in many of which the detailed structure of the constituting units and their interactions may not be known to a degree of certainty comparable to that of a physical law) are captured in this way through models governed by simple local rules. In their computer implementation these models provide attractive visualizations and deep insights, from Monte Carlo and multi-agent simulations to cellular automata and games.

## Applications

Inspiration from complex systems is being applied to gain understanding on large scale natural systems such as the atmosphere and climate, starting from first principles, notably in connection with the crucial issue of prediction. Conversely, natural complexity acts as a source of inspiration for progress at the fundamental level.

Complex systems are at the origin of new techniques for artificial self-organizing and computational devices in such contexts as biotechnology, information science and robotics, where decentralized interactions of simple autonomous units lead to swarm intelligence and global structures complementary to those of conventional machines, able to respond and adapt with minimal outside direction, robust to damage and highly flexible. Evolutionary principles stemming from complexity research also play a role in the understanding of the functioning of the immune system and the brain as well as in the development of artificial neural networks and related systems capable of performing pattern recognition, optimization, etc.

Complex systems constitute a privileged interface between mathematical and physical sciences on the one side, and social and economic sciences on the other. Here, while the laws governing the evolution are not known to any comparable degree of detail as in a physico-chemical system, many of the behaviors observed are part of the characteristic phenomenology of complex systems as outlined in this article. It is thus natural to take advantage of the concepts and the techniques elaborated in the context of a physically-based complexity theory to tackle some of the problems arising in these disciplines from an interesting angle. In doing so, one is often led to proceed by analogy. Market dynamics, management, transport, decision making, are among the most promising directions bringing, in addition, the distinctive feature of being composed of individual elements with internal adaptation and response. This introduces a new level of complexity associated with the co-evolution of the components with each other and with their external environment.

## References

**Surveys**

- Complex Systems, Nature Insight 410, 241-284 (2001).
- S. Camazine, J.-L. Deneubourg, N. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-organization in biological systems, Princeton University Press, Princeton (2001).
- H. Haken, Synergetics. An introduction: Nonequilibrium phase transitions and self-organization in physics, chemistry and biology, Springer, Berlin (1977).
- H. Haken, Advanced Synergetics. Instability hierarchies and self-organizing systems and devices, Springer, Berlin (1983).
- J.H. Holland, Emergence : from chaos to order, Helix Books, New York (1998).
- S. Kauffman, The origins of order, Oxford University Press, New York (1993).
- G. Nicolis and C. Nicolis, Foundations of Complex Systems, World Scientific, Singapore (2007).
- G. Nicolis and I. Prigogine, Self-organization in nonequilibrium systems, Wiley, New York (1977).
- S. Wolfram, A new kind of Science, Wolfram Media Inc., Champaign, Illinois (2002).\n\n\n

**Specific topics**

- R. Albert and A.L. Barabasi, Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47-97 (2002).
- P.W. Anderson, K.J. Arrow and D. Pines (eds), The economy as an evolving complex system, Addison-Wesley, Redwood City, CA (1988).
- G. Chaitin, MetaMath? The quest for Omega, http://arxiv.org/abs/math.-HO/0404335.
- A. Hartmann and H. Rieger (eds), New optimization algorithms in physics, Wiley, New York (2004).
- B. Mandelbrot, Fractales, hasard et finance, Flammarion, Paris (1997).
- C. Nicolis and G. Nicolis, Chaos in dissipative systems : Understanding atmospheric physics, Adv. Chem. Phys. 91, 511-570 (1995).
- I. Prigogine, The end of certainty, The Free Press, New York (1997).
- A. Scott (ed.), Encyclopedia of nonlinear science, Taylor and Francis, New York (2005).
- Whole-istic Biology, Science 295, 1661-1682 (2002).

**Internal references**

- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- Edward Ott (2006) Basin of attraction. Scholarpedia, 1(8):1701.
- Anatol M. Zhabotinsky (2007) Belousov-Zhabotinsky reaction. Scholarpedia, 2(9):1435.
- John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
- Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.
- Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
- Edward Ott (2006) Crises. Scholarpedia, 1(10):1700.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Henrik Jeldtoft Jensen and Paolo Sibani (2007) Glassy dynamics. Scholarpedia, 2(6):2030.
- James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
- Marco Dorigo and Mauro Birattari (2007) Swarm intelligence. Scholarpedia, 2(9):1462.
- Hermann Haken (2007) Synergetics. Scholarpedia, 2(1):1400.

## See also

Algorithmic Complexity, Attractor, Basin of Attraction, Bifurcations, Cellular Automata, Chaos, Complexity, Dynamical Systems, Entropy, Fractals, Glassy Dynamics, Kolmogorov-Sinai Entropy, Normal Forms, Self-organization, Synergetics, Turbulence