It is known that inverse cascades arise in discontinuous logistic maps [de Sousa Viera and co-workers, Phys. Rev. A 35, 945 (1987); in Universalities in Condensed Matter, edited by R. Jullien, L. Peliti, R. Rammal, and N. Boccara (Springer, New York, 1988); in Disordered Systems and Biological Models, edited by L. Peliti (World Scientific, Singapore, 1989); Europhys. Lett. 9, 119 (1989)]. Here we report on the discovery of direct cascades in such maps. Between any two consecutive terms of an inverse cascade there may or may not occur new cycles. We have discovered two empirical rules for the existence of these new cycles. When they do exist, the periods of the new cycles can be predicted by a rule we have discovered numerically, namely the summation rule, which involves a method of generating terms identical to the way terms of a Farey sequence are generated. By application of the summation rule, it is possible to generate both higher-level direct and inverse cascades which have self-similar structures. Further, the existence of direct cascades can also be explained by the same rule. We discuss the role of the discontinuity of the map in relation to bifurcations. We also mention how the cascades and the summation rule become modified in the chaotic region of the map.

• Received 8 January 1992

DOI:https://doi.org/10.1103/PhysRevA.45.8441