We consider the generalized Korteweg - de Vries (KdV) equation with periodic boundary condition by space variable. The asymptotics of periodic solutions and invariant tori are considered in a sufficiently small neighbourhood of the zero equilibrium state. There are considered two cases: the coefficient of the quadratic term in the right part of generalized KdV equation is non-zero or zero. In the first case, the cycles and two-dimension tori are unstable. In the second, case it is proved that cycles are stable and tori are unstable.