Cyclic Structure of Dynamical Systems Associated with 3 x + d Extensions of Collatz Problem

We study here, from both theoretical and experimental points of view, the cyclic structures, both general and primitive, of dynamical systems Dd generated by iterations of the functions Td acting, for all d ≥ 1 relatively prime to 6, on positive integers : Td : N −→ N; Td(n) = { n 2 , if n is even; 3n+d 2 , if n is odd. In the case d = 1, the properties of the system D = D1 are the subject of the well-known 3x + 1 conjecture. For every one of 6667 systems Dd, 1 ≤ d ≤ 19999, we calculate its (complete, as we argue) list of primitive cycles. We unite in a single conceptual framework of primitive memberships, and we experimentally confirm three primitive cycles conjectures of Jeff Lagarias. An in-deep analysis of the diophantine formulae for primitive cycles, together with new rich experimental data, suggest several new conjectures, theoretically studied and experimentally confirmed in the present paper. As a part of this program, we prove a new upper bound to the number of primitive cycles of a given oddlength.