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Cylindrical dust structures formed by single-size grains and by two-size grains (small and large) are considered. The confinement perpendicular to the cylinder axis is due either to an external parabolic potential or to a self-consistent field of smaller grain structure. Dense small grain structures can produce confinement for a few of the larger grains which can form a helical structure. Two methods, the molecular dynamic (MD) simulations and the analytical numerical calculation of the equilibrium states, are used to find the dust structure configurations. The interactions between the grains is considered as the sum of: 1) either Yukawa screened potential Vscr∝exp(-r/λYuk) (valid for small grain charges) or non-Yukawa nonlinear screened potential Vscr∝(1/r)(1-r/λnscr)4 (valid for grain charges of typical existing experiments); 2) the nonscreened noncollective attraction (Vncol∝-(1/r); and 3) the nonscreened collective attraction Vcol∝(1/r)cos(r/λcol) (in presence of the fields created by small grains). The MD simulations use the dust neutral gas friction. It is shown that for any random initial cylindrical distributions of grains and for all types of interactions, the grains reach an equilibrium state as a helical structure with a constant rotation angle. With an increase of the radius of the structure, bifurcations are observed converting a single spiral to a double spiral, then to a triple spiral, etc. An increase of the structure radius leads to a sudden bifurcation of the rotation angle for N-winding structures within ranges of fixed N. The number of bifurcations increases with a decrease of the confining constant, with an increase of the structure radius, and with an increase of the attraction interactions. A complete set of bifurcations is found for different interacting potentials. Results of analytical methods used for numerical calculations of equilibrium structures are in agreement with the results obtained in MD simulations. In the presence of collective or noncollective attraction, the stable helical structures can exist with a negative or zero external confining constant. The latter presents a boundary-free helical- structure. An equation for small perturbations of the equilibrium helical structure state is found and the numerical solution of it is used to investigate the frequencies of the helical modes. The calculated mode frequencies and the bifurcations of the rotation angle can serve as a tool for measurements of the interdust interactions and as "memory marks" in helical structures. These structures can be created in laboratory experiments and can be present in space. Such self-organized "memorizing" dust structures, if they are present in nature, should create modulations of the space infrared emission.