Related papers: Center manifolds without a phase space
We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse…
We extend the theory of spectral submanifolds (SSMs) to general non-autonomous dynamical systems that are either weakly forced or slowly varying. Examples of such systems arise in structural dynamics, fluid-structure interactions and…
The theory of slow manifolds is an important tool in the study of deterministic dynamical systems, giving a practical method by which to reduce the number of relevant degrees of freedom in a model, thereby often resulting in a considerable…
Noise-induced transitions between metastable fixed points in systems evolving on multiple time scales are analyzed in situations where the time scale separation gives rise to a slow manifold with bifurcation. This analysis is performed…
We propose a framework for a global description of the dynamics of complex flows via clusterized spatial representations of the flow, isolating and identifying local dynamics, retrieving different Space-Time Cluster-Based Network Models…
As Neural Cellular Automata (NCAs) are increasingly applied outside of the toy models in Artificial Life, there is a pressing need to understand how they behave and to build appropriate routes to interpret what they have learnt. By their…
It is well known that conservative mechanical systems exhibit local oscillatory behaviours due to their elastic and gravitational potentials, which completely characterise these periodic motions together with the inertial properties of the…
(Completely regular) locales generalize (Tychonoff) spaces; indeed, the passage from a locale to its spatial sublocale is a well understood coreflection. But a locale also possesses an equally important pointless sublocale, and with…
We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform…
Modular flows probe important aspects of the entanglement structures, especially those of QFTs, in a dynamical framework. Despite the expected non-local nature in the general cases, the majority of explicitly understood examples feature…
We construct a sheaf theoretic and derived geometric machinery to study nonlinear partial differential equations and their singular supports. We establish a notion of derived microlocalization for solution spaces of non-linear equations and…
The problem of decomposing non-manifold object has already been studied in solid modeling. However, the few proposed solutions are limited to the problem of decomposing solids described through their boundaries. In this thesis we study the…
This work proposes an autoencoder neural network as a non-linear generalization of projection-based methods for solving Partial Differential Equations (PDEs). The proposed deep learning architecture presented is capable of generating the…
We study local bifurcations of periodic solutions to time-periodic (systems of) integrodifference equations over compact habitats. Such infinite-dimensional discrete dynamical systems arise in theoretical ecology as models to describe the…
The reduction of Hamiltonian systems aims to build smaller reduced models, valid over a certain range of time and parameters, in order to reduce computing time. By maintaining the Hamiltonian structure in the reduced model, certain…
Spectral subspaces of a linear dynamical system identify a large class of invariant structures that highlight/isolate the dynamics associated to select subsets of the spectrum. The corresponding notion for nonlinear systems is that of…
We survey a (nonlinear) Fredholm theory for a new class of ambient spaces called polyfolds, and develop the analytical foundations for some of the applications of the theory. The basic feature of these new spaces, which can be finite and…
The communications and interrelations between different locations on the Earth's surface have far-reaching implications for both social and natural systems. Effective spatial analytics ideally require a spatial representation, where…
This work develops the global equations of neural networks through stacked piecewise manifolds, fixed-point theory, and boundary-conditioned iteration. Once fixed coordinates and operators are removed, a neural network appears as a…
Subspace clustering is an important unsupervised clustering approach. It is based on the assumption that the high-dimensional data points are approximately distributed around several low-dimensional linear subspaces. The majority of the…