Sporadicity: Between periodic and chaotic dynamical behaviors.

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 --> infinity. 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.