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The dynamical systems herein are Morse-Smale diffeomorphisms and flows on      C ∞  {C^\infty }       compact manifolds. We show the asymptotic rate of approach of orbits to the sinks of the systems to be bounded by an expression of the form      K exp ⁡  ( −  D N ) K\;\exp ( - DN)      , where      D D       may be any number smaller than      C = min p { 1 / m log Jac D P f m | W u ( P ) } C = {\min _p}\{ 1/m\;\log \;\operatorname {Jac}\;{D_P}\,{f^m}|{W^u}(P)\}      . Here the minimum is taken over all nonsink      P P       in the nonwandering set of      f f      , and      m m       is the period of      P P      . We extend our theorems to the entire manifold, so that there is no restriction on the location of the initial points of trajectories.

Rate of approach to minima and sinks—the Morse-Smale case