Related papers: Analytic continuation and differential geometry vi…
We consider discrete dynamical systems and lattice models in statistical mechanics from the point of view of their symmetry groups. We describe a C program for symmetry analysis of discrete systems. Among other features, the program…
This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a…
Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of…
The dynamics of perfect fluid spacetime geometries which exhibit {\em Local Rotational Symmetry} (LRS) are reformulated in the language of a $1+\,3$ "threading" decomposition of the spacetime manifold, where covariant fluid and curvature…
Chemical reactions modeled by ordinary differential equations are finite-dimensional dissipative dynamical systems with multiple time-scales. They are numerically hard to tackle -- especially when they enter an optimal control problem as…
This article deals with invariant manifolds for infinite dimensional random dynamical systems with different time scales. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations. Under suitable…
We introduce a general differentiable solver for time-dependent deformation problems with contact and friction. Our approach uses a finite element discretization with a high-order time integrator coupled with the recently proposed…
This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation…
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…
We provide a compact derivation of the static and dynamic equations for infinite-dimensional particle systems in the liquid and glass phases. The static derivation is based on the introduction of an "auxiliary" disorder and the use of the…
A dynamical system with discrete time is studied by means of algebraic geometry. The system admits a reduction that is interpreted as a classical field theory in 2+1-dimensional wholly discrete space-time. The integrals of motion of a…
The basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated, leading to the conclusion that if they are differential equations on its phase space $M$, they are at most of the 2nd order.…
A new field of numerical astrophysics is introduced which addresses the solution of large, multidimensional structural or slowly-evolving problems (rotating stars, interacting binaries, thick advective accretion disks, four dimensional…
The Method of Invariant Grid (MIG) is an iterative procedure for model reduction in chemical kinetics which is based on the notion of Slow Invariant Manifold (SIM) [1-4]. Important role, in that method, is played by the initial grid which,…
We develop Riemannian approaches to variational autoencoders (VAEs) for PDE-type ambient data with regularizing geometric latent dynamics, which we refer to as VAE-DLM, or VAEs with dynamical latent manifolds. We redevelop the VAE framework…
We develop a unified spectral-semigroup framework that connects real-time and imaginary-time quantum dynamics through analytic continuation. Within this formulation, evolution is expressed as an exponential reweighting of spectral…
I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of…
We show how spectral submanifold theory can be used to provide analytic predictions for the response of periodically forced multi-degree-of-freedom mechanical systems. These predictions include an explicit criterion for the existence of…
A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves…
We propose a method for learning dynamical systems from high-dimensional empirical data that combines variational autoencoders and (spatio-)temporal attention within a framework designed to enforce certain scientifically-motivated…