Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation

Tese de doutoramento. Ciências da Engenharia. 2006. Faculdade de Engenharia. Universidade do Porto, Instituto Superior Técnico. Universidade Técnica de Lisbo

Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation

We present an algorithm for the rigorous integration of Delay Differential Equations (DDEs) of the form $x'(t)=f(x(t-\tau),x(t))$. As an application, we give a computer assisted proof of the existence of two attracting periodic orbits (before and after the first period-doubling bifurcation) in the Mackey-Glass equation

Is "the theory of everything'' merely the ultimate ensemble theory?

We discuss some physical consequences of what might be called ``the ultimate ensemble theory'', where not only worlds corresponding to say different sets of initial data or different physical constants are considered equally real, but also worlds ruled by altogether different equations. The only postulate in this theory is that all structures that exist mathematically exist also physically, by which we mean that in those complex enough to contain self-aware substructures (SASs), these SASs will subjectively perceive themselves as existing in a physically ``real'' world. We find that it is far from clear that this simple theory, which has no free parameters whatsoever, is observationally ruled out. The predictions of the theory take the form of probability distributions for the outcome of experiments, which makes it testable. In addition, it may be possible to rule it out by comparing its a priori predictions for the observable attributes of nature (the particle masses, the dimensionality of spacetime, etc) with what is observed.Comment: 29 pages, revised to match version published in Annals of Physics. The New Scientist article and color figures are available at http://www.sns.ias.edu/~max/toe_frames.html or from [email protected]

Spectral theory of damped quantum chaotic systems

We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on $X$ and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also present a new condition for a spectral gap, depending on the set of minimally damped trajectories.Comment: Lecture given at the Journ\'ees semiclassiques 2011, Biarritz, 6-10 June 201

Efficient method for detection of periodic orbits in chaotic maps and flows

An algorithm for detecting unstable periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp. 6172-6175] which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp. 4733-4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighbouring periods, has been shown to be highly efficient when applied to low-dimensional system. The difficulty in applying the algorithm to higher dimensional systems is mainly due to the fact that the number of stabilising transformations grows extremely fast with increasing system dimension. In this thesis, we construct stabilising transformations based on the knowledge of the stability matrices of already detected periodic orbits (used as seeds). The advantage of our approach is in a substantial reduction of the number of transformations, which increases the efficiency of the detection algorithm, especially in the case of high-dimensional systems. The performance of the new approach is illustrated by its application to the four-dimensional kicked double rotor map, a six-dimensional system of three coupled H\'enon maps and to the Kuramoto-Sivashinsky system in the weakly turbulent regime.Comment: PhD thesis, 119 pages. Due to restrictions on the size of files uploaded, some of the figures are of rather poor quality. If necessary a quality copy may be obtained (approximately 1MB in pdf) by emailing me at [email protected]

A Conley index study of the evolution of the Lorenz strange set

In this paper we study the Lorenz equations using the perspective of the Conley index theory. More specifically, we examine the evolution of the strange set that these equations posses throughout the different values of the parameter. We also analyze some natural Morse decompositions of the global attractor of the system and the role of the strange set in these decompositions. We calculate the corresponding Morse equations and study their change along the successive bifurcations. In addition, we formulate and prove some theorems which are applicable in more general situations. These theorems refer to Poincar\'{e}-Andronov-Hopf bifurcations of arbitrary codimension, bifurcations with two homoclinic loops and a study of the role of the travelling repellers in the transformation of repeller-attractor pairs into attractor-repeller ones.Comment: 22 pages, 1 figur

Reduced order modeling of convection-dominated flows, dimensionality reduction and stabilization

We present methodologies for reduced order modeling of convection dominated flows. Accordingly, three main problems are addressed. Firstly, an optimal manifold is realized to enhance reducibility of convection dominated flows. We design a low-rank auto-encoder to specifically reduce the dimensionality of solution arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Also, considering that the latent variables are often hard to interpret, many of these methods are dismissed in the reduced order modeling of dynamical systems governed by partial differential equations (PDEs). This deficiency is of importance to the extent that linear methods, such as principle component analysis (PCA) and Koopman operators, are still prevalent. Accordingly, we propose an interpretable nonlinear dimensionality reduction algorithm. An unsupervised learning problem is constructed that learns a diffeomorphic spatio-temporal grid which registers the output sequence of the PDEs on a non-uniform time-varying grid. The Kolmogorov n-width of the mapped data on the learned grid is minimized. Secondly, the reduced order models are constructed on the realized manifolds. We project the high fidelity models on the learned manifold, leading to a time-varying system of equations. Moreover, as a data-driven model free architecture, recurrent neural networks on the learned manifold are trained, showing versatility of the proposed framework. Finally, a stabilization method is developed to maintain stability and accuracy of the projection based ROMs on the learned manifold a posteriori. We extend the eigenvalue reassignment method of stabilization of linear time-invariant ROMs, to the more general case of linear time-varying systems. Through a post-processing step, the ROMs are controlled using a constrained nonlinear lease-square minimization problem. The controller and the input signals are defined at the algebraic level, using left and right singular vectors of the reduced system matrices. The proposed stabilization method is general and applicable to a large variety of linear time-varying ROMs