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Complex Systems: Introduction

Computational Complexity



Complex systems cover a wide range of phenomena and disciplines with many partially overlapping definitions of complexity (the common characteristic of complex systems). These definitions manage to capture important aspects of our common intuitions about complex phenomena, but up to now scientists have been unable to reach a universal definition that would do justice to these perspectives and join them in an all-embracing formulation.

One way to grasp the concept is to contrast complex systems with regular, non-complex ones. Simple (or non-complex) systems have been the most frequent subjects of study for traditional physics from the times of Galileo until roughly the 1960’s (planetary systems, hydraulic pumps, electric circuits, etc.). Non-complex systems can be modeled as sets of simple basic parts that interact weakly or not at all with each other and evolve according to especially simple (linear) dynamic laws. Some characteristic features often include:

  • linearity
  • time-reversibility
  • stability
  • determinism

By contrast, complex systems are characterized by, among other features, the presence of:

  • non-linearity
  • irreversibility
  • indeterminism

Other typical characteristics that will be considered separately are:

  • emergence
  • self-organization
  • symmetry breaking

Another feature of complex systems is that they tend to display unusual combinations of ordered and disordered phenomena.

Typical examples of complex systems are:

  • living beings and their associations (tissues, organs, ecosystems, colonies, bacterial films, etc.)
  • far-from-equilibrium thermodynamic systems* (e.g., a tornado)
  • phase transitions* (e.g., boiling or freezing of water) and critical phenomena* in condensed matter physics (e.g., ferromagnetism, superconductivity, etc.)
  • street traffic patterns
  • financial markets
  • pattern formation and collective motion (flocking of birds, etc.)

Complexity research radically transcends traditional barriers between established disciplines and actively demands collaborative interdisciplinary research and the trans-disciplinary generalization of methods and concepts from the most diverse branches of knowledge and practices.

The Illusion of Complexity - Neuroactivity and Complex Behavior

"The Illusion of Complexity - Neuroactivity and Complex Behavior" (7:04).  More information at