Related papers: Analytic continuation and differential geometry vi…
A delayed term in a differential equation reflects the fact that information takes significant time to travel from one place to another within a process being studied. Despite de apparent similarity with ordinary differential equations,…
The disadvantage of `traditional' multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis…
Lubrication equations allow to describe many structurin processes of thin liquid films. We develop and apply numerical tools suitable for their analysis employing a dynamical systems approach. In particular, we present a time integration…
The manifold hypothesis suggests that high-dimensional neural time series lie on a low-dimensional manifold shaped by simpler underlying dynamics. To uncover this structure, latent dynamical variable models such as state-space models,…
To obtain explicit understanding of the behavior of dynamical systems, geometrical methods and slow-fast analysis have proved to be highly useful. Such methods are standard for smooth dynamical systems, and increasingly used for continuous,…
A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism…
We study a class of slow-fast Hamiltonian systems with any finite number of degrees of freedom, but with at least one slow one and two fast ones. At $% \epsilon =0$ the slow dynamics is frozen. We assume that the frozen system (i.e. the…
The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this…
If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as…
We investigate the geometric, algebraic and homologic structures related with Poisson structure on a smooth manifold. Introduce a noncommutative foundations of these structures for a Poisson algebra. Introduce and investigate noncommutative…
We study dynamic heterogeneities in a model glass-former whose overlap with a reference configuration is constrained to a fixed value. The system phase-separates into regions of small and large overlap, so that dynamical correlations remain…
In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent…
We apply topological methods to obtain global continuation results for harmonic solutions of some periodically perturbed ordinary differential equations on a $k$-dimensional differentiable manifold $M \subseteq \mathbb{R}^m$. We assume that…
This paper presents the continuous and discrete variational formulations of simple thermodynamical systems whose configuration space is a (finite dimensional) Lie group. We follow the variational approach to nonequilibrium thermodynamics…
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral…
Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state…
We extend Cellular Automata to time-varying discrete geometries. In other words we formalize, and prove theorems about, the intuitive idea of a discrete manifold which evolves in time, subject to two natural constraints: the evolution does…
Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} $X\subset C^1([-h,0],\mathbb{R}^n)$. For systems with discrete…
We present an explicit multiscale algorithm for solving differential equations for problems with high-frequency modes that can be averaged over by separating and scaling the fast and slow dynamics within a single equation. We introduce a…
This paper describes the notion of \sigma -symmetry, which extends the one of \lambda-symmetry, and its application to reduction procedures of systems of ordinary differential equations and of dynamical systems as well. We also consider…