The lyapunov exponent is a measure for the exponential divergence of a small perturbations in initial conditions of a dynamical system. A positive exponent means that a small difference in initial conditions will cause exponentially big differences after enough time has passed. A butterfly flapping it's wings and changing the wind speeds in a cubic decimeter of air in one place may eventually lead to a huge storm on the other side of the world. A system with a positive exponent is chaotic.

The lyapunov exponent is defined as the limit for time goes to infinity of the natural logarithm of the growth of an infinitessimal offset in initial conditions.

A simple example of a system with a positive lyapunov exponent is the map of 2x modulo 1. The lyapunov exponent for this map is ln(2).

In general the initial conditions may be perturbed in many different ways and therefore a system has as many lyapunov exponents as it's phase space has dimensions. Some may be equal to eachother.

Systems which are time reversible, such as a system with colliding hard spheres, have a negative exponent for every positive one. This is called the conjugate pairing rule. Under time reversal every positive exponent becomes a negative one. The system remains the same and therefore must have the same lyapunov exponents. Similar properties hold for some not time reversible systems.