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Astron. Astrophys. 343, 325-332 (1999)

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Appendix A:

The Lyapunov characteristic exponent [FORMULA] of two nearby trajectories, [FORMULA] and [FORMULA], is defined by

[EQUATION]

where [FORMULA] is a solution of the equations of motion in the phase space, [FORMULA] is a connecting vector, [FORMULA] is its length in the phase space and [FORMULA] is an initial separation (the value of [FORMULA] must be small; otherwise it is arbitrary and one usually scales [FORMULA] to unity). The Kerr spacetime being stationary, we can measure the separations on the [FORMULA] surfaces. One defines [FORMULA] as the maximum value of [FORMULA] with respect to variations of w and characterizes the chaoticity of the system by the value of [FORMULA]. Positive values of [FORMULA] indicate that neighbouring trajectories diverge exponentially in the course of time while negative [FORMULA] corresponds only to polynomial divergence. The above definitions have been originally introduced within the framework of non-relativistic systems but they are directly applicable also to stationary systems in general relativity.

The maximum Lyapunov characteristic exponent is frequently determined numerically. This approach requires a careful choice of the integration scheme. In order to keep computational errors under control, one lets the trajectories evolve for a short interval of time, [FORMULA], after which w is rescaled back to unity (one denotes [FORMULA] the norm of the connecting vector at the moment of the k-th rescaling). One can show (Benettin 1984) that the Lyapunov characteristic exponent corresponding to the original [FORMULA] is given by

[EQUATION]

independently of the value of [FORMULA] In addition, [FORMULA] for almost all [FORMULA] Again, one can conveniently set [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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