In the present paper stationary pure-exchange overlapping generations economies with l  goods per date and m consumers per generation are considered. It is shown that for an open and dense set of utility functions there exist endowment vectors such that n-cycles exist for n = l +1 and l  = m. The approach to existence of endogenous fluctuations is basic in the sense that the prime ingredients are the implicit function theorem and linear algebra. Moreover the approach is applied to show that for an open and dense set of utility functions there exist endowment vectors such that sunspot equilibria, where prices at every date only depends on the state at that date, exist.