This thesis studies the qualitative theory of linear and nonlinear infinite dimensional dynamical systems with applications mainly to parabolic partial differential equations. The objective of the study is to examine through linearization the local and global behaviour, including existence and nonexistence, of invariant structures such as equilibria and periodic solutions.;In the linear theory, the dimension of the asymptotically stable solution subspace of a linear differential equation is studied. This gives new insights into the behaviour of linear and nonlinear dynamical systems.;The nonlinear results include such topics as a generalization to infinite dimensional differential equations of a classical stability condition of Poincare. The main idea is that a periodic orbit is stable if the system diminishes nearby 2-dimensional areas. Similar considerations give conditions for the existence as well as the stability of a periodic solution. If the system diminishes areas globally rather than locally, it is shown that nontrivial periodic solutions can not exist; this is a generalization of the well-known 2-dimensional Bendixson condition for the nonexistence of periodic solutions.;Examples of applications to concrete differential equations are given throughout and the thesis concludes with an application of the Bendixson condition to an epidemiological model.
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