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>Computer-assisted studies and visualization of nonlinear phenomena: Two-dimensional invariant manifolds, global bifurcations, and robustness of global attractors.
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Computer-assisted studies and visualization of nonlinear phenomena: Two-dimensional invariant manifolds, global bifurcations, and robustness of global attractors.
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.
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机译：计算机图形学和数值算法与分析工具一起用于研究一维周期性Kuramoto-Sivashinsky方程（KSE）的长期行为{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}几种开发健壮算法来计算两个的方法研究了平衡点和鞍型极限环的三维不变流形。交互式算法和可视化为研究全局分叉附近相空间的质变提供了有效的手段。使用这些工具，对两个系统的现象学研究的结果（三阶微分方程和KSE的近似惯性形式）以一系列系统参数上多个（不稳定）歧管的相互作用表示西尔尼科夫同宿分叉附近。 KSE的研究以一个猜想结束，该猜想将周期解的拓扑性质联系在一起，并且在相空间中远离这些周期解的稳态之间创建了无限多的异斜率连接。本论文的最后一章致力于定量了解全球吸引子的稳健性。对于固定的，周期性的，正的，偶数的函数{dollar} alpha = alpha（x）> 0，{dollar} KSE（0.1）的空间扰动{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研究了{lcub} x {rcub} rbrack = 0，等式（0.2）{美元} {美元}。特别地，通过限制为奇数u和偶数{dollar} alpha {dollar}，我们证明了该方程是耗散的；此外，对于较大的值，渐近地获得了{spall} Lsp2（0，2pi {dollar}）和{dollar} Hsp1（0，2pi {dollar}）中吸收球的半径以及整体吸引子的尺寸的估计值。 {dollar} alpha（x {dollar}）及其衍生工具的各种规范。最后，使用频谱屏障方法显示该系统具有惯性流形，其尺寸也已估算。
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