Related papers: Delayed Pattern Formation in Two-Dimensional Domai…
We investigate the energetic ground states of a model two-phase system with 1/r^3 dipolar interactions in two dimensions. The model exhibits spontaneous formation of two kinds of periodic domain structure. A striped domain structure is…
To generate coherent responses, language models infer unobserved meaning from their input text sequence. One potential explanation for this capability arises from theories of delay embeddings in dynamical systems, which prove that…
The growth of domains of stripes evolving from random initial conditions is studied in numerical simulations of models of systems far from equilibrium such as Rayleigh-Benard convection. The scaling of the size of the domains deduced from…
High-dimensional multivariate spatial-temporal data arise frequently in a wide range of applications; however, there are relatively few statistical methods that can simultaneously deal with spatial, temporal and variable-wise dependencies…
Pattern formation from homogeneity is well-studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is nontrivial to separate observed spatial patterning due to inherent spatial heterogeneity from…
Pattern dynamics triggered by fixing a boundary is investigated. By considering a reaction-diffusion equation that has a unique spatially-uniform and limit cycle attractor under a periodic or Neumann boundary condition, and then by choosing…
Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. To understand the latent space $\mathbf{x}_t \in \mathcal{X}$, we analyze them from a geometrical perspective. Our approach…
Spatiotemporal patterns are common in biological systems. For electrically-coupled cells previous studies of pattern formation have mainly used external forcing as the main bifurcation parameter. The purpose of this paper is to show that…
Networks of interconnected agents are essential to study complex networked systems' state evolution, stability, resilience, and control. Nevertheless, the high dimensionality and nonlinear dynamics are vital factors preventing us from…
In pattern-forming systems, competition between patterns with different wave numbers can lead to domain structures, which consist of regions with differing wave numbers separated by domain walls. For domain structures well above threshold…
In order to explore the impact of periodically evolving domain on the transmission of disease, we study a SIS reaction-diffusion model with logistic term on a periodically evolving domain. The basic reproduction number ${\mathcal{R}}_0$ is…
The denaturation dynamics of a long double-stranded DNA is studied by means of a model of the Poland-Scheraga type. We note that the linking of the two strands is a locally conserved quantity, hence we introduce local updates that respect…
Time series forecasting is vital in many real-world applications, yet developing models that generalize well on unseen relevant domains -- such as forecasting web traffic data on new platforms/websites or estimating e-commerce demand in new…
Normal and anomalous diffusion are ubiquitous in many complex systems [1] . Here, we define a time and space generalized diffusion equation (GDE), which uses fractional-time derivatives and transformed d-path Laplacian operators on…
A class of hyperbolic reaction--diffusion models with cross-diffusion is derived within the context of Extended Thermodynamics. Linear stability analysis is performed to study the nature of the equilibrium states against uniform and…
Neurons are often connected, spatially and temporally, in phenomenal ways that promote wave propagation. Therefore, it is essential to analyze the emergent spatiotemporal patterns to understand the working mechanism of brain activity,…
Driven by the explosion of data and the impact of real-world networks, a wide array of mathematical models have been proposed to understand the structure and evolution of such systems, especially in the temporal context. Recent advances in…
In this work an activator-depleted reaction-diffusion system is investigated on polar coordinates with the aim of exploring the relationship and the corresponding influence of domain size on the types of possible diffusion-driven…
Various degenerate diffusion equations exhibit a waiting time phenomenon: Dependening on the "flatness" of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount…
In this article we investigate the semiflow properties of a class of state-dependent delay differential equations which is motivated by some models describing the dynamics of the number of adult trees in forests. We investigate the…