**5. Conclusion**

The following chapter covers five year work that has been conducted in the Chaotic Dynamics Laboratory in the Institute for Systems Analysis of Russian Academy of Sci., lead by professor N.A. Magnitskii. Our attempt in using standard open source or commercial software for this kind of analysis failed so we had to consider a specially constructed accurate and trustworthy numerical solution code for Navier-Stokes equations, partly described here. The results of numerical solution for initial-boundary value problems considered confirmed that laminar-turbulent transition undergoing the bifurcation process and has different scenarios. It is interesting to point out that similar scenarios with classical Feigenbaum scenario and Sharkovskiy windows of periodicity where recently found in [66] for initial-boundary value problems in continuous mechanical systems such as flexible plates and shallow shells. However, the universal FSM scenario [7] is found in all problems considered, despite the difference between problems. Recently we found that Boltzmann equations in hydrodynamic limit with BGK collision integral [63–65] also exhibit FSM scenario for laminar-turbulent transition process. It is likely that all hydrodynamic type chaotic solutions for PDEs are undergoing the FSM scenario in various modifications. The work continues and now we are considering compressible fluid dynamics (transonic and supersonic turbulence) and magnetohydrodynamics as well as other initial-boundary value problems for (1).
