  In this work, we study the long time behaviors, including asymptotic
contractivity and dissipativity, of the solutions to several numerical methods
for fractional ordinary differential equations (F-ODEs). The existing algebraic
contractivity and dissipativity rates of the solutions to the scalar F-ODEs are
first improved. In order to study the long time behavior of numerical solutions
to fractional backward differential formulas (F-BDFs), two crucial analytical
techniques are developed, with the first one for the discrete version of the
fractional generalization of the traditional Leibniz rule, and the other for
the algebraic decay rate of the solution to a linear Volterra difference
equation. By mens of these auxiliary tools and some natural conditions, the
solutions to F-BDFs are shown to be contractive and dissipative, and also
preserve the exact contractivity rate of the continuous solutions. Two typical
F-BDFs, based on the Grunwald-Letnikov formula and L1 method respectively, are
studied. For high order F-BDFs, including some second order F-BDFs and
$3-\alpha$ order method, their numerical contractivity and dissipativity are
also developed under some slightly stronger conditions. Numerical experiments
are presented to validate the long time qualitative characteristics of the
solutions to F-BDFs, revealing very different decay rates of the numerical
solutions in terms of the the initial values between F-ODEs and integer ODEs
and demonstrating the superiority of the structure-preserving numerical
methods.
