We define the class of sporadic dynamical systems as the systems where the algorithmic complexity of Kolmogorov [Kolmogorov, A. N. (1983) Russ. Math. Surv. 38, 29-40] and Chaitin [Chaitin, G. J. (1987) Algorithmic Information Theory (Cambridge Univ. Press, Cambridge, U.K.)] as well as the logarithm of separation of initially nearby trajectories grow as n v 0 (log n ) v 1 with 0 < v 0 < 1 or v 0 = 1 and v 1 < 0 as time n → ∞. These systems present a behavior intermediate between the multiperiodic ( v 0 = 0, v 1 = 1) and the chaotic ones ( v 0 = 1, v 1 = 0). We show that intermittent systems of Manneville [Manneville, P. (1980) J. Phys. (Paris) 41, 1235-1243] as well as some countable Markov chains may be sporadic and, furthermore, that the dynamical fluctuations of these systems may be of Lévy's type rather than Gaussian.