Abstract
For most values of xd ∈ ( - 1,1), the logistic map with a sectional discontinuity at x = xd possesses at least one inverse cascade. By using the property that, when xd is positive or zero, every first cascade accumulates at a parameter a = aacc immediately at the end of a 2-cycle, we explain the functional dependence of aacc on xd. Further, we derive hitherto unknown, general analytical expressions for aacc when xd lies in the range (0,0.9); in particular, these expressions give values of aacc identical to those previously found by a computational technique for selected values of xd in the same range [B. L. Tan and T. T Chia, Phys. Rev. E 47, 3087 (1993)]. We also present a method for calculating the values of the bifurcation points within any inverse cascade for this map and for the TB map which consists of two piecewise linear portions [A. S. Lima, I. C. Moreira, and A. M. Serra, Phys. Lett. A 190, 403 (1994)].
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