  We consider the relation between relaxation time and the largest Lyapunov
exponent in a system of two coupled oscillators, one of them being harmonic. It
has been found that in a rather broad region of parameter space, contrary to
the common expectation, both Lyapunov exponent and relaxation time increase as
a function of the total energy. This effect is attributed to the fact that
above a critical value of the total energy, although the Lyapunov exponent
increases, KAM tori appear and the chaotic fraction of phase space decreases.
We examine the required conditions and demonstrate the key role of the
dispersion relation for this behaviour to occur. This study is useful, among
others, in the understanding of the damping of nuclear giant resonances.
