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We discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order. The differential operators are taken in the Riemann–Liouville sense and the initial conditions are specified according to Caputo's suggestion, thus allowing for interpretation in a physically meaningful way. We investigate in particular the dependence of the solution on the order of the differential equation and on the initial condition, and we relate our results to the selection of appropriate numerical schemes for the solution of fractional differential equations.