Why do we refer to Lyapunov exponents as a characteristic of the system, telling us about its chaoticity, when it is only referred to a point?

Question

Lyapunov exponents are defined as indicators of the sensibility to initial conditions. In fact, they give the mean rate of exponential separation of trajectories. They are, however, specific of a given point, in which the Jacobian matrix that defines the exponent is indeed calculated, so I don't understand why we say that the fact that the exponent is positive can give us an information regarding the whole system, which we can say is therefore chaotic.

Answer 1

One way to think about this is that even though Lyapunov exponents of a system (say the iterates of a diffeomorphism or the flow of a first order ODE) are defined with reference to a basepoint at first, they can be considered as (partially defined) functions defined on the phase space $M$ with $d$ dimensions. Moreover, the classical Oseledets theorem guarantees that there is a subset $M_0$ of the phase space that is full measure with respect to any Borel probability measure invariant under the time evolution such that for any basepoint $x\in M_0$ Lyapunov exponents $\chi^1_x,\chi^2_x,...,\chi^{d}_x\in\mathbb{R}$ (possibly with multiplicity) are well-defined, and indeed uniquely defined by the system. Furthermore, considered as functions they are Borel measurable, and they are invariant under the time evolution.

For diffeomorphisms or flows a classical theorem by Krylov-Bogoliubov guarantees the existence of invariant Borel probability measures, and each such measure has a so-called "ergodic decomposition". With respect to such an ergodic measure one can take the integrals of the Lyapunov exponents (which are invariant measurable functions) to obtain numerical Lyapunov exponents (which are now attached to not only the dynamical system but to the dynamical system paired with a certain "optimal" way of observing it). In this way one can also keep the system fixed and vary the ergodic invariant measure to think of the Lyapunov exponents of the systems as functions that associate to each ergodic measure (one "optimal" way of observing) how surprising the observed behavior is.

Thus either by thinking of Lyapunov exponents as functions defined on the phase space, or as functions defined on the space of invariant ergodic measures, one can take them as global objects.

Answer 2

Your premise is not correct. When people speak of the Lyapunov exponent, they usually mean the largest Lyapunov exponent averaged along a given trajectory, such as a limit cycle or a chaotic attractor.

There are also local Lyapunov exponents, but even those are more intricate to be computed simply from the Jacobian of the dynamics.