Lyapunov Characteristic Exponent

  In section 2.10 we informally introduced the Lyapunov exponent as a simple measure of sensitive dependence on initial conditions, i.e., chaotic  behavior. The notion of a Lyapunov exponent is a generalization of the idea of an eigenvalue as a measure of the stability of a fixed point or a characteristic exponent [1] as the measure of the stability of a periodic orbit. For a chaotic trajectory it is not sensible to examine the instantaneous eigenvalue of a trajectory. The next best quantity, therefore, is an eigenvalue averaged over the whole trajectory. The idea of measuring the average stability of a trajectory leads us to the formal notion of a Lyapunov exponent. The Lyapunov exponent is best defined by looking at the evolution (under a flow) of the tangent manifold. That is, ``sensitive dependence on initial conditions'' is most clearly stated as an observation about the evolution of vectors in the tangent manifold rather than the evolution of trajectories in the flow of the original manifold M.

  
Figure 4.24: Evolution of vectors in the tangent manifold under a flow.

The tangent manifold  at a point x, written as , is the collection of all tangent vectors of the manifold M at the point x. The tangent manifold is a linear vector space. The collection of all tangent manifolds is called the tangent bundle . For instance, for a surface embedded in the tangent manifold at each point of the manifold is a tangent plane. More generally, if the original manifold is of dimension n, then the tangent manifold is a linear vector space also of dimension n. For further background material on manifolds, tangent manifolds, and flows, see Arnold [1].

The integral curves of a flow on a manifold provide a smooth foliation of that manifold, in the following manner. A point goes to the point under the flow (see Fig. 4.24). Now we make a key observation: the tangent vectors are also carried by the flow (this is called a ``Lie dragging'') so that we can set up a unique correspondence between the tangent vectors in and the tangent vectors in . Namely, for each , there exists a unique vector . The Lyapunov characteristic exponent  of a flow is defined as

 

That is, the Lyapunov characteristic exponent measures the average growth rate of vectors in the tangent manifold. The corresponding Lyapunov characteristic exponent of a map is [6]

 

where

 

A flow is said to have sensitive dependence on initial conditions  if the Lyapunov characteristic exponent is positive. From a physical point of view, the Lyapunov exponent is a very useful indicator distinguishing a chaotic  from a nonchaotic trajectory [16].