Formation and development of patterns in various phenomena from fluid dynamics to biological organs is a very fascinating subject. The issues of phase transition and pattern formation, in mechanisms governed by a dynamical system, requires a rigorous bifurcation analysis. This appears to be one of the very challenging questions in theoretical studies.

The aim of this work is to study bifurcations and transitions of two famous dynamical systems, namely the Swift-Hohenberg equation and the inhibitor-activator model. The bifurcation analysis is performed in a framework of attractor bifurcation theory which aims to identify and classify the local basin of attractions of a given system near critical values. A main aspect of the present investigation is the precise center manifold reduction and derivation of reduced equations in each particular case.

Chapter one is a survey of the key concepts necessary for the bifurcation analysis, such as the notion of attractor bifurcation, the abstract bifurcation theorem, and the center manifold reduction method. Chapter two concerns the bifurcation and asymptotic behavior of solutions of the Swift-Hohenberg equation and the generalized Swift-Hohenberg equation. In chapter three a system of reaction and diffusion of two morphogens is reduced to a finite system of ordinary differential equations, and bifurcation and asymptotic behavior of bifurcated objects are analyzed.