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The methods of the bifurcation theory of codimension two, together with computer calculations, are used to investigate stationary, periodic, and quasiperiodic flows with two and three independent frequencies, as well as chaotic regimes of fluid flow between two infinite rotating permeable concentric cylinders near the intersection of the bifurcations initiating secondary stationary flow and self-oscillations with azimuthal waves.The experimental investigations [1][2][3] have shown that with increase in the Reynolds number the main stationary rotational-symmetric fluid flow between two rotating permeable cylinders changes for a secondary stationary flow or a self-oscillatory regime with waves traveling in the azimuthal direction. The further increase in the Reynolds number leads to complication of the fluid flow structure and gives rise to different complicated regimes and then to turbulence.The calculations of neutral curves made it possible to found that at certain parameter values the curves corresponding to rotational-symmetric and oscillatory three-dimensional instabilities intersect [4][5][6][7][8].In the mid-eighties of the last century V.I. Yudovich in Russia and J. Iooss and P. Chossat in France devised the bifurcation theory of codimension two for hydrodynamic flows with cylindrical symmetries. This made it possible to investigate different fluid flow regimes in the vicinity of the point of intersection of bifurcations of the origin of secondary stationary flow and azimuthal waves for impermeable cylinders [9,10]. In this study, this theory is applied for calculating complicated fluid flows in the Couette-Taylor problem for permeable cylinders. GOVERNING EQUATIONS AND MAIN REGIMELet a gap between two solid infinite permeable concentric cylinders, R 1 and R 2 in radii (R 1 < R 2 ) rotating at angular velocities Ω 1 and Ω 2 be filled with a viscous homogeneous incompressible fluid. We will assume that there are no external body forces. The quantities R 1 , Ω 1 R 1 ,a nd 1/Ω 1 are taken for the length, velocity, and time scales, respectively.In the cylindrical coordinates r, ϕ, z, with the z axis aligned with the cylinder axis, the dimensionless Navier-Stokes and continuity equations take the form:

This paper considers viscous incompressible fluid flows between two infinite rotating permeable concentric cylinders near the bifurcation point, which result in secondary steady flow and oscillations with azimuthal waves. Steady periodic and quasiperiodic fluid flow regimes with two, three, and four independent frequencies are obtained by methods of the theory of codimension-two bifurcations of hydrodynamic flows having cylindrical symmetry.

Abstract-The method is based on identifications of a particular solution of the linear dynamic system on a class of static models with the dynamic specification on an input. On the basis of a particular solution the common decision are search. The common decision is a basis for an estimation of a spectrum system. For identification of a spectrum eigenvalues we introduced special structures, who described a modification of Lyapunov exponents. We gave generalization of the offer methods on the linear non-stationary dynamic systems. The algorithm of decision-making about a type of eigenvalues system is developed. We analyse properties of the offered special functions.Index Terms-Identification, lyapunov exponent, spectrum of eigenvalues, structure.

Ability of plants to purify contaminated indoor air using vertical hardware-biological complex of recirculating type was studied. Laboratory experiments were made for assessing capability to purify indoor air against different particle size of contaminated air by cigarettes. Comparative analysis of the data obtained with the use of Tradescantia fluminensis Vell and Sansevieria trifasciata was shown. According to the results of laboratory tests and diagnostic status of the air, making the following options by location of vertical hardware-biological complex of recirculating type.