Related papers: Spatial discretizations of generic dynamical syste…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…
We present a method to determine the shot noise in quantum systems from knowledge of their time evolution - the latter being obtained using numerical simulation techniques. While our ultimate goal is the study of interacting systems, the…
The study addresses matter-coupled modified gravity, particularly $f (T, \mathcal{T})$ gravity, unveiling distinct formalism. The research further discusses stability analysis, and the dynamical system approach, exploring the dynamics of…
We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach…
We approach the discretisation of Maxwell's equations directly in space-time without making any non-relativistic assumptions with the particular focus on simulations in rotating reference frames. As a research example we study Sagnac's…
In this paper, we provide sufficient conditions for dissipativity and local asymptotic stability of discrete-time dynamical systems parametrized by deep neural networks. We leverage the representation of neural networks as pointwise affine…
Nonlinear PDEs give rise to complex dynamics that are often difficult to analyze in state space due to their relatively large numbers of degrees of freedom, ill-conditioned operators, and changing spatial and parameter resolution…
The relation between the results of cosmological N-body simulations, and the continuum theoretical models they simulate, is currently not understood in a way which allows a quantification of N dependent effects. In this first of a series of…
The structure of functional graphs of nonlinear systems provides one of the most intuitive methods for analyzing their properties in digital domain. The generalized Tent map is particularly suitable for studying the degradation of dynamic…
A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…
In this paper we provide optimal bounds for fully discrete approximations to finite horizon problems via dynamic programming. We adapt the error analysis in \cite{nos} for the infinite horizon case to the finite horizon case. We prove an a…
While the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with $ N $ variables, as binary neural networks and cellular automata. The main difficulty is the…
We can approximate a continuous self-map $f$ of a compact metric space by discretizing the space into a grid. Through either the map itself or a time series, $f$ induces a multivalued grid map $\mathcal F$. The dynamical properties of…
There are many classical results, related to the Denjoy--Wolff Theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, for the…
Rare trajectories of stochastic systems are important to understand -- because of their potential impact. However, their properties are by definition difficult to sample directly. Population dynamics provides a numerical tool allowing their…
The use of numerical simulation for prediction of characteristics of chaotic dynamical systems inherently involves unpredictable processes. In this work, we develop a model for the expected error in the simulation of ergodic, chaotic ODE…
In this paper, we present and analyze a new finite difference method for computing three dimensional wave maps into spheres. By introducing the angular momentum as an auxiliary variable, we recast the governing equation as a first order…
Grid-based neural networks such as convolutional autoencoders are widely used in dimension reduction-based surrogate models for computational fluid dynamics. In recent years, the use of coordinate-based approaches like conditional neural…
Computer simulation of the time evolution in a classical system is a standard numerical method, used in numerous scientific articles in Natural Science. Almost all the simulations are performed by discrete Molecular Dynamics (MD). The…
Understanding dissipative and decohering processes is fundamental to the study of quantum systems. An accurate and generic method for investigating these processes is to simulate both the system and environment, which, however, is…