See Complexity theory for definition and theory. See Complex systems for examples and applications.
Complexity is a general concept that has different meanings in different contexts. For instance, complexity is related to “incompressibility” in information theory and computer science. In dynamical systems, complexity is usually measured by the topological entropy and reflects roughly speaking, the proliferation of periodic orbits with ever longer periods or the number of orbits that can be distinguished with increasing precision. In physics, the label “complex” is in principle attached to any nonlinear system whose numerical solutions exhibit a chaotic behavior. Neurologists claim that the human brain is the most complex system in the solar system, while entomologists teach us the baffling complexity of some insect societies. The list could be enlarged with examples from geometry, management science, commu- nication and social networks, etc.
from book on Permutation complexity by Amigo
Complexity can be seen in chaotic systems, and, depending on interpretation, in random ones. It is also important in biological systems (see Biological complexity).
See also Randomness
Universal Measures of Complexity - Algorithmic Nature Group http://complexitycalculator.com/
Good review: RANDOMNESS, INFORMATION, AND COMPLEXITY
Information and Complexity Measures in Dynamical Systems
See also Information theory, Statistical physics, Dynamical systems, Evolution, Simplicity bias.
Complexity ~ Modular hierarchical structures.
Information measures, effective complexity, and total information