| Computer graphics and numerical algorithms are used in conjunction with analytical tools in the study of long-time behavior of the one-dimensional periodic Kuramoto-Sivashinsky equation (KSE){dollar}{dollar}usb{lcub}t{rcub}+usb{lcub}xxxx{rcub}+alphalbrack usb{lcub}xx{rcub}+uusb{lcub}x{rcub}rbrack=0.eqno(0.1){dollar}{dollar}Several approaches toward the development of robust algorithms to compute two-dimensional invariant manifolds of equilibrium points and saddle-type limit cycles are studied. Interactive algorithms and visualization provide an effective means by which to study qualitative changes of phase space near global bifurcations.; Using these tools, the results of a phenomenological study of two systems--a third order differential equation and an approximate inertial form of the KSE--are expressed in terms of the interplay of multiple (un)stable manifolds over a range of system parameters near Silnikov homoclinic bifurcations. The KSE study concludes with a conjecture connecting topological properties of periodic solutions and the creation of infinitely-many heteroclinic connections between steady states far away from these periodic solutions in phase space.; The final chapter of this Thesis is devoted toward a quantitative understanding of the robustness of global attractors. For a fixed, periodic, positive, even function {dollar}alpha = alpha(x) > 0,{dollar} a spatial perturbation of the KSE (0.1){dollar}{dollar}usb{lcub}t{rcub} + usb{lcub}xxxx{rcub} + (alpha usb{lcub}x{rcub})sb{lcub}x{rcub} + {lcub}1over3{rcub}lbrackalpha uusb{lcub}x{rcub} + (alpha usp2)sb{lcub}x{rcub}rbrack = 0,eqno(0.2){dollar}{dollar}is studied. Particularly, by restricting to odd u and even {dollar}alpha{dollar}, we prove that this equation is dissipative; furthermore, estimates of the radius of the absorbing ball in {dollar}Lsp2(0, 2pi{dollar}) and {dollar}Hsp1(0, 2pi{dollar}) and to the dimension of the global attractor are obtained asymptotically for large values of various norms of {dollar}alpha(x{dollar}) and its derivatives. Lastly, using the method of spectral barriers, this system is shown to possess an inertial manifold, the dimension of which is also estimated. |
