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Using both numerical and analytical approaches, we demonstrate the existenceof an effective power-law relation $L\propto m^p$ between the mean Lyapunovexponent $L$ of stellar orbits chaotically scattered by a supermassive blackhole in the center of a galaxy and the mass parameter $m$, i.e. ratio of themass of the black hole over the mass of the galaxy. The exponent $p$ is foundnumerically to obtain values in the range $p \approx 0.3$--$0.5$. We propose atheoretical interpretation of these exponents, based on estimates of local`stretching numbers', i.e. local Lyapunov exponents at successive transits ofthe 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 ellipticalgalaxy models with a central core. However, we find numerically that aneffective power law scaling of $L$ with $m$ holds also in models with centralcusp, beyond a mass scale up to which chaos is dominated by the influence ofthe cusp itself. We finally show numerically that an analogous law exists alsoin disc galaxies with rotating bars. In the latter case, chaotic scattering bythe black hole affects mainly populations of thick tube-like orbits surroundingsome low-order branches of the $x_1$ family of periodic orbits, as well as itsbifurcations at low-order resonances, mainly the Inner Lindbland resonance andthe 4/1 resonance. Implications of the correlations between $L$ and $m$ todetermining the rate of secular evolution of galaxies are discussed.

Effective power-law dependence of Lyapunov exponents on the central mass in galaxies