A cocycle refers to any one of the following:

• A closed cochain in algebraic topology
  Algebraic topology
  Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
  
   is called a cocycle.
• A particular type of map
  Function (mathematics)
  In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
  
   in an autonomous dynamical system
  Dynamical system
  A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
  
  ; see Oseledec theorem
  Oseledec theorem
  In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets in 1965 and reported at the International Mathematical Congress in Moscow in 1966...
  
  .

Let G be a graph with vertex set V. A cut is a partition S = {X, X'} of V into two nonempty subsets. We denote the set of all edges incident with one vertex in X and one vertex in X' by E(S) or E(X, X'); and any such edge set is called a cocycle. [pg 103 ISBN 1431-1550]