Related papers: Global Topology from an Embedding
We describe a class of fractal attractors induced by \beta-shifts. We use a coding by these shifts to show that the systems are mixing with topological entropy log \beta and have an ergodic measure of full entropy. Moreover we determine the…
A vector topology on a vector space over a topological field is a (not necessarily Hausdorff) topology by which the addition and scalar multiplication are continuous. We prove that, if an isomorphism between the lattice of topologies of two…
The systems without symmetries, e.g. the spatial and chiral symmetries, are generally thought to be improper for topological study and no conventional integral topological invariant can be well defined. In this work, with multi-band…
The notion of an attractor has various definitions in the theory of dynamical systems. Under compactness assumptions, several of those definitions coincide and the theory is rather complete. However, without compactness, the picture becomes…
Multiplex networks are collections of networks with identical nodes but distinct layers of edges. They are genuine representations for a large variety of real systems whose elements interact in multiple fashions or flavors. However,…
Dimensional evolution between one- ($1D$) and two-dimensional ($2D$) topological phases is investigated systematically. The crossover from a $2D$ topological insulator to its $1D$ limit shows oscillating behavior between a $1D$ ordinary…
A generalization of the Lorenz equations is proposed where the variables take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can…
We present a fully automated method for the optimal state space reconstruction from univariate and multivariate time series. The proposed methodology generalizes the time delay embedding procedure by unifying two promising ideas in a…
A generalized attracting horseshoe is introduced as a new paradigm for describing chaotic strange attractors (of arbitrary finite rank) for smooth and piecewise smooth maps f from Q to Q, where Q is a homeomorph of the unit interval in real…
The paper deals with topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finitedimensional smooth systems can exist in three different forms.…
In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an…
We analyze a phase-field system where the energy balance equation is linearly coupled with a nonlinear and nonlocal ODE for the order parameter $\chi$. The latter equation is characterized by a space convolution term which models particle…
How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use…
We consider the pattern formation problem in coupled identical systems after the global synchronized state becomes unstable. Based on analytical results relating the coupling strengths and the instability of each spatial mode (pattern) we…
Long term behavior of nonlinear deterministic continuous time signals can be studied in terms of their reconstructed attractors. Reconstructed attractors of a continuous signal are meant to be topologically equivalent representations of the…
Considering random noise in finite dimensional parameterized families of diffeomorphisms of a compact finite dimensional boundaryless manifold M, we show the existence of time averages for almost every orbit of each point of M, imposing…
This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits a weakly attracting…
The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system is comprised of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of…
Clustering $\unicode{x2013}$ the tendency for neighbors of nodes to be connected $\unicode{x2013}$ quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase…
Inspired by the generation power of generative adversarial networks (GANs) in image domains, we introduce a novel hierarchical architecture for learning characteristic topological features from a single arbitrary input graph via GANs. The…