相关论文: Topological synchronisation or a simple attractor?

We study the relationship between the partial synchronous (PaS) state and the coupling structure in general dynamical systems. By the exact proof, we find the sufficient and necessary condition of the existence of PaS state for the coupling…

We generalize various notions of stability of invariant sets of dynamical systems to invariant measures, by defining a topology on the set of measures. The defined topology is similar, but not topologically equivalent to weak* topology, and…

Robustness against small perturbations is a crucial feature of topological properties. This robustness is both a source of theoretical interest and a drive for technological applications, but presents a challenge when looking for new…

Synchronization of identical harmonic oscillators interconnected via position, velocity, and acceleration couplings is studied. How to construct a complex Laplacian matrix representing the overall coupling is presented. It is shown that the…

Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in…

An interesting problem in synchronization is the study of coupled oscillators, wherein oscillators with different natural frequencies synchronize to a common frequency and equilibrium phase difference. In this paper, we investigate the…

Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…

We introduce a new method for determining the global stability of synchronization in systems of coupled identical maps. The method is based on the study of invariant measures. Besides the simplest non-trivial example, namely two…

Topological Machine Learning (TML) is an emerging field that leverages techniques from algebraic topology to analyze complex data structures in ways that traditional machine learning methods may not capture. This tutorial provides a…

This paper introduces the induced matching distance, a novel topological metric designed to compare discrete structures represented by a symmetric non-negative function. We apply this notion to analyze agent trajectories over time. We use…

In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible…

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…

Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a…

Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…

Synchronization of coupled continuous-time linear systems is studied in a general setting. For identical neutrally-stable linear systems that are detectable from their outputs, it is shown that a linear output feedback law exists under…

The celebrated Takens' embedding theorem concerns embedding an attractor of a dynamical system in a Euclidean space of appropriate dimension through a generic delay-observation map. The embedding also establishes a topological conjugacy. In…

The problem of a harmonic oscillator coupling to an electromagnetic potential plus a topological-like (Chern-Simons) massive term, in two-dimensional space, is studied in the light of the symplectic formalism proposed by Faddeev and Jackiw…

Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (non-linear in general) functionals that are linear on singly…

We study the relationship between topological scales and dynamic time scales in complex networks. The analysis is based on the full dynamics towards synchronization of a system of coupled oscillators. In the synchronization process, modular…

We obtain sufficient conditions for the stability of the synchronized solutions for a class of coupled dynamical systems. This is accomplished by finding an analytical expression for the transverse Liapunov exponent through spectral…