Related papers: Random templex encodes topological tipping points …
We study dynamical systems forced by a combination of random and deterministic noise and provide criteria, in terms of Lyapunov exponents, for the existence of random attractors with continuous structure in the fibres. For this purpose, we…
A cyclic random walk is a random walk whose transition probabilities/rates can be written as a superposition of the empirical measures of a family of finite cycles. This identifies a convex set of models. We discuss the problem of…
When a low dimensional chaotic attractor is embedded in a three dimensional space its topological properties are embedding-dependent. We show that there are just three topological properties that depend on the embedding: parity, global…
Annealed functional CLT in the rough path topology is proved for the standard class of ballistic random walks in random environment. Moreover, the `area anomaly', i.e. a deterministic linear correction for the second level iterated integral…
A broad range of nonlinear processes over networks are governed by threshold dynamics. So far, existing mathematical theory characterizing the behavior of such systems has largely been concerned with the case where the thresholds are…
Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…
We present a neurobiologically--inspired stochastic cellular automaton whose state jumps with time between the attractors corresponding to a series of stored patterns. The jumping varies from regular to chaotic as the model parameters are…
The emergence of noise-induced chaos in a random logistic map with bounded noise is understood as a two-step process consisting of a topological bifurcation flagged by a zero-crossing point of the supremum of the dichotomy spectrum and a…
Most algorithms for representation learning and link prediction on relational data are designed for static data. However, the data to which they are applied typically evolves over time, including online social networks or interactions…
We set up and analyze a random matrix model to study energy localization and its time behavior in two chaotically coupled systems. This investigation is prompted by a recent experimental and theoretical study of Weaver and Lobkis on coupled…
We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the…
Identifying ordered structures hidden in the packings of particles is a common scientific question in multiple fields. In this work, we investigate the dynamical organizations of a large number of initially randomly packed repulsive…
Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and…
Motivated by the idea of using simple macroscopic examples to illustrate the physics of complex systems, we modify a historic experimental setup in which interacting floating magnets spontaneously self-assemble into ordered clusters. By…
Tipping is a phenomenon in multistable systems where small changes in inputs cause huge changes in outputs. When the parameter varies within a certain time scale, the rate will affect the tipping behaviors. These behaviors are undesirable…
In neural circuits, statistical connectivity rules strongly depend on neuronal type. Here we study dynamics of neural networks with cell-type specific connectivity by extending the dynamic mean field method, and find that these networks…
We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called "thermodynamic" regime. We prove the existence of universal limit laws for the topologies; namely, the…
Algorithms for node clustering typically focus on finding homophilous structure in graphs. That is, they find sets of similar nodes with many edges within, rather than across, the clusters. However, graphs often also exhibit heterophilous…
Topological features in quantum computing provide controllability and noise error avoidance in the performance of logical gates. While such resilience is favored in the manipulation of quantum systems, it is very hard to identify…
Continuous and discrete time systems possessing strange non-chaotic attractors are under investigation. It is demonstrated that unpredictable trajectories exist in the dynamics. A recent numerical technique, the sequential test, is utilized…