The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the Maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate, the effect of the other exponents will be obliterated over time.
One may look into the following:
[1] H. González-Aguilar and E. Ugalde (eds): Nonlinear Dynamics New Directions: Theoretical Aspects (Springer, Switzerland, 2015).
[2] M. Lakshamanan and S. Rajasekhar: Nonlinear Dynamics: Integrability, Chaos, and Patterns (Springer, New York, 2002).
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