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We show that the Lyapunov exponents of a periodic orbit can be easilyobtained from the eigenvalues of the monodromy matrix. It turns out that theLyapunov exponents of simply stable periodic orbits are all zero, simplyunstable periodic orbits have only one positive Lyapunov exponent, doublyunstable periodic orbits have two different positive Lyapunov exponents and thetwo positive Lyapunov exponents of complex unstable periodic orbits are equal.We present a numerical example for periodic orbits in a realistic galacticpotential. Moreover, the center manifold theorem allowed us to show thatstable, simply unstable and doubly unstable periodic orbits are the mothers offamilies of, respectively, regular, partially and fully chaotic orbits in theirneighbourhood.

The Lyapunov exponents and the neighbourhood of periodic orbits