Related papers: Dimensional-invariance principles in coupled dynam…
One of the key challenges in the dimension theory of smooth dynamical systems is in establishing whether or not the Hausdorff, lower and upper box dimensions coincide for invariant sets. For sets invariant under conformal dynamics, these…
Consider a $k$-valued network. Two kinds of (control) invariant subspaces, called state and dual invariant subspaces, are proposed, which are subspaces of state space and dual space respectively. Algorithms are presented to verify whether a…
To preserve previously learned representations, continual learning systems must strike a balance between plasticity, the ability to acquire new knowledge, and stability. This stability-plasticity dilemma affects how representations can be…
Dynamic mode decomposition (DMD) provides a principled approach to extract physically interpretable spatial modes from time-resolved flow field data, along with a linear model for how the amplitudes of these modes evolve in time. Recently,…
Global aspects of the motion of passive scalars in time-dependent incompressible fluid flows are well described by volume-preserving (Liouvillian) three-dimensional maps. In this paper the possible invariant structures in Liouvillian maps…
We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems, which we call dynamical dimension reduction (DDR). In the DDR model, each point is evolved via a nonlinear flow towards…
We analyze the linear stability of monoclinal traveling waves on a constant incline, which connect uniform flowing regions of differing depths. The classical shallow-water equations are employed, subject to a general resistive drag term.…
Memristive systems, namely resistive systems with memory, are attracting considerable attention due to their ubiquity in several phenomena and technological applications. Here, we show that even the simplest one-dimensional network formed…
Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the…
The dynamical systems found in Nature are rarely isolated. Instead they interact and influence each other. The coupling functions that connect them contain detailed information about the functional mechanisms underlying the interactions and…
Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincar\'e first made this connection by tracking consecutive…
This paper focuses on multiscale dynamics occurring in steam supply systems. The dynamics of interest are originally described by a distributed-parameter model for fast steam flows over a pipe network coupled with a lumped-parameter model…
Dynamical behaviour of discrete dynamical systems has been investigated extensively in the past few decades. However, in several applications, long term memory plays an important role in the evolution of dynamical variables. The definition…
Recent work on stochastic interacting particle systems with two particle species (or single-species systems with kinematic constraints) has demonstrated the existence of spontaneous symmetry breaking, long-range order and phase coexistence…
Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in…
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…
In this paper, we provide an overview of the research conducted in the context of structural systems since the latest survey by Dion et al. in 2003. We systematically consider all the papers that cite this survey as well as the seminal work…
Convection is a well-studied topic in fluid dynamics, yet it is less understood in the context of networks flows. Here, we incorporate techniques from topological data analysis (namely, persistent homology) to automate the detection and…
We provide an overview of high dimensional dynamical systems driven by random matrices, focusing on applications to simple models of learning and generalization in machine learning theory. Using both cavity method arguments and path…
All interesting and fascinating collective properties of a complex system arise from the intricate way in which its components interact. Various systems in physics, biology, social sciences and engineering have been successfully modelled as…