We study here, from both theoretical and experimental points of view, the cyclic structures, both general and primitive, of dynamical systems ${cal D}_d$ generated by iterations of the functions $T_d$ acting, for all $dgeq 1$ relatively prime to 6, on positive integers : $$T_d : {f N} longrightarrow {f N}; qquad T_d(n) = cases{hskip 0.6em elax {n over 2} &, if $n$ is even; {3n+d over 2} &, if $n$ is odd. cr}$$ In the case $d = 1$, the properties of the system ${cal D} = {cal D}_1$ are the subject of the well-known $3x+1$ conjecture. For every one of 6667 systems ${cal D}_d, 1le d le 19999$, 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.