Effective power-law dependence of Lyapunov exponents on the central mass
  in galaxies
  Using both numerical and analytical approaches, we demonstrate the existence
of an effective power-law relation $L\propto m^p$ between the mean Lyapunov
exponent $L$ of stellar orbits chaotically scattered by a supermassive black
hole in the center of a galaxy and the mass parameter $m$, i.e. ratio of the
mass of the black hole over the mass of the galaxy. The exponent $p$ is found
numerically to obtain values in the range $p \approx 0.3$--$0.5$. We propose a
theoretical interpretation of these exponents, based on estimates of local
`stretching numbers', i.e. local Lyapunov exponents at successive transits of
the orbits through the black hole's sphere of influence. We thus predict
$p=2/3-q$ with $q\approx 0.1$--$0.2$. Our basic model refers to elliptical
galaxy models with a central core. However, we find numerically that an
effective power law scaling of $L$ with $m$ holds also in models with central
cusp, beyond a mass scale up to which chaos is dominated by the influence of
the cusp itself. We finally show numerically that an analogous law exists also
in disc galaxies with rotating bars. In the latter case, chaotic scattering by
the black hole affects mainly populations of thick tube-like orbits surrounding
some low-order branches of the $x_1$ family of periodic orbits, as well as its
bifurcations at low-order resonances, mainly the Inner Lindbland resonance and
the 4/1 resonance. Implications of the correlations between $L$ and $m$ to
determining the rate of secular evolution of galaxies are discussed.
