A new construction of regular solutions to the three dimensional Navier{Stokes equa- tions is introduced and applied to the study of their asymptotic expansions. This construction and other Phragmen-Linderl??of type estimates are used to establish su??- cient conditions for the convergence of those expansions. The construction also de??nes a system of inhomogeneous di??erential equations, called the extended Navier{Stokes equations, which turns out to have global solutions in suitably constructed normed spaces. Moreover, in these spaces, the normal form of the Navier{Stokes equations associated with the terms of the asymptotic expansions is a well-behaved in??nite system of di??erential equations. An application of those asymptotic expansions of regular solutions is the analysis of the helicity for large times. The dichotomy of the helicity's asymptotic behavior is then established. Furthermore, the relations between the helicity and the energy in several cases are described.

<p>Ova doktorska disertacija uvodi nov pristup ispitivanju stabilnosti dinamičkih<br />sistema, korišćenjem teorije pseudospektra. Na taj način se postojeći pojam<br />stabilnosti profinjuje pojmom robusne stabilnosti, koji mnogo adekvatnije<br />opisuje realnu ekološku stabilnost. Razvijen je nov matematički alat za<br />izračunavanje indikatora stabilnosti, koji je zatim ilustrovan na primeru dva<br />ekosistema tla, sa po četiri uzorka, u četiri različita stadijuma razvoja.</p> / <p>This doctoral dissertation establishes a novel approach to the stability analysis of<br />dynamical systems, in terms of matrix pseudospectrum. In that manner, the existing<br />concept of stability has undergone essential refinement so as to give birth to the<br />concept of robust stability, which has the ability to capture the ecological stability at a<br />more adequate level. Additionally, within the framework of the dissertation, a new<br />mathematical tool for the stability indicators computation has been developed, which<br />has then been used to illustrate theoretical results in form of two soil ecosystems,<br />each of them sampled four times, all of them observed in four distinct stages of<br />evolution.</p>