Complex system

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This article largely discusses complex systems as a subject of mathematics and the attempts to emulate physical complex systems with emergent properties. For other scientific and professional disciplines addressing complexity in their fields see the complex systems article and references.

A complex system is a damped, driven system (for example, a harmonic oscillator) whose total energy exceeds the threshold for it to perform according to classical mechanics but does not reach the threshold for the system to exhibit properties according to chaos theory.


Although it is arguable that humans have been studying complex systems for thousands of years, the modern scientific study of complex systems is relatively young in comparison to conventional fields of science with simple system assumptions, such as physics and chemistry. The history of the scientific study of these systems follows several different research trends.

In the area of mathematics, arguably the largest contribution to the study of complex systems was the discovery of chaos in deterministic systems, a feature of certain dynamical systems that is strongly related to nonlinearity.[1] The study of neural networks was also integral in advancing the mathematics needed to study complex systems.


Types of complex systems[edit]

Chaotic systems[edit]

For a dynamical system to be classified as chaotic, it must have the following properties:[2]

  1. it must be sensitive to initial conditions,
  2. it must be topologically mixing, and
  3. its periodic orbits must be dense.


Complex adaptive systems[edit]


Nonlinear system[edit]

The behaviour of non-linear systems is not subject to the principle of superposition while that of linear systems is subject to superposition. Thus, a complex nonlinear system is one whose behaviour cannot be expressed as a sum of the behaviour of its parts (or of their multiples). [3]

Topics on complex systems[edit]

Features of complex systems[edit]

Complex systems may have the following features:

Cascading Failures
Complex systems may be open
Complex systems may have a memory
Complex systems may be nested
Dynamic network of multiplicity
As well as coupling rules, the dynamic network of a complex system is important. Small-world or scale-free networks[5][6][7] which have many local interactions and a smaller number of inter-area connections are often employed. Natural complex systems often exhibit such topologies. In the human cortex for example, we see dense local connectivity and a few very long axon projections between regions inside the cortex and to other brain regions.
May produce emergent phenomena
Relationships are non-linear
Relationships contain feedback loops

See also[edit]


  1. ^ History of Complex Systems
  3. ^ EPSRC description of Non-linear systems retrieved 11 Aug 2015
  4. ^ S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin (2010). "Catastrophic cascade of failures in interdependent networks". Nature 464 (7291): 08932. doi:10.1038/nature08932. PMID 20393559. 
  5. ^ A. L. Barab´asi, R. Albert (2002). "Statistical mechanics of complex networks". Rev. Mod. Phys 74: 47–94. Bibcode:2002RvMP...74...47A. doi:10.1103/RevModPhys.74.47. 
  6. ^ M. Newman (2010). Networks: An Introduction. Oxford University Press. ISBN 978-0-19-920665-0. 
  7. ^ Reuven Cohen, Shlomo Havlin (2010). Complex Networks: Structure, Robustness and Function. Cambridge University Press. ISBN 978-0-521-84156-6. 

Further reading[edit]

External links[edit]