Structural complexity is a science of
applied mathematics, that aims at relating fundamental physical or biological aspects of a
complex system
A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication ...
with the mathematical description of the morphological complexity that the system exhibits, by establishing rigorous relations between mathematical and physical properties of such system.
Structural complexity emerges from all systems that display morphological organization. Filamentary structures, for instance, are an example of
coherent structures that emerge, interact and evolve in many physical and biological systems, such as mass distribution in the
Universe,
vortex filaments in turbulent flows,
neural networks
A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
in our brain and genetic material (such as
DNA) in a cell. In general information on the degree of morphological
disorder present in the system tells us something important about fundamental physical or biological processes.
Structural complexity methods are based on applications of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
topology (and in particular
knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
) to interpret physical properties of
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
. such as relations between
kinetic energy and tangles of vortex filaments in a turbulent flow or
magnetic energy and braiding of magnetic fields in the solar corona, including aspects of
topological fluid dynamics.
Literature
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References
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Applied mathematics
Complex systems theory