Related papers: Constructive Methods of Invariant Manifolds for Ki…
In this chapter we summarize recent developments in the study of kinetically constrained models (KCMs) as models for glass formers. After recalling the definition of the KCMs which we cover we study the possible occurrence of ergodicity…
This paper presents an approach to systematically design sliding mode control and manifold to stabilize nonlinear uncertain systems. The objective is also accomplished to enlarge the inner bound of region of attraction for closed-loop…
Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $\Delta_g$ on…
The one-dimensional system of equations of isentropic gas dynamics is considered. First-order invariants of characteristics of this system are classified. Second-order invariants of characteristics are classified for polytropic processes.…
We undertake a systematic exploration of recurrent patterns in a 1-dimensional Kuramoto-Sivashinsky system. For a small, but already rather turbulent system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set…
We present a simple technique for the computation of coarse-scale steady states of dynamical systems with time scale separation in the form of a "wrapper" around a fine-scale simulator. We discuss how this approach alleviates certain…
In this work we propose a novel method to ensure important entropy inequalities are satisfied semi-discretely when constructing reduced order models (ROMs) on nonlinear reduced manifolds. We are in particular interested in ROMs of systems…
Coupled wave equations are popular tool for investigating longitudinal dynamical effects in semiconductor lasers, for example, sensitivity to delayed optical feedback. We study a model that consists of a hyperbolic linear system of partial…
We obtain global and local theorems on the existence of invariant manifolds for perturbations of non autonomous linear differential equations assuming a very general form of dichotomic behavior for the linear equation. Besides some new…
Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering. The existence of a strange attractor in the turbulent…
We present a data-driven and interpretable approach for reducing the dimensionality of chaotic systems using spectral submanifolds (SSMs). Emanating from fixed points or periodic orbits, these SSMs are low-dimensional inertial manifolds…
In the present paper we give a brief summary of some recent theoretical advances in the treatment of inhomogeneous fluids and methods which have applications in the study of dynamical properties of liquids in situations of extreme…
We consider the development of high order asymptotic-preserving linear multistep methods for kinetic equations and related problems. The methods are first developed for BGK-like kinetic models and then extended to the case of the full…
We present a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically…
We present a recipe for rendering a submanifold normally hyperbolic and invariant within a stability basin. The construction includes the ability to choose the asymptotic phase map. We are motivated by the notion of "templates and anchors"…
In this note we provide conditions for local invariance of finite dimensional submanifolds for solutions to stochastic partial differential equations (SPDEs) in the framework of the variational approach. For this purpose, we provide a…
We provide general methods for explicitly constructing strict Lyapunov functions for fully nonlinear slowly time-varying systems. Our results apply to cases where the given dynamics and corresponding frozen dynamics are not necessarily…
Multiscale stochastic dynamical systems have been widely adopted to a variety of scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to…
In this article, we review the progress made on the statistical mechanics of liquids and fluids embedded in curved space. Our main focus will be on two-dimensional manifolds of constant nonzero curvature and on the influence of the latter…
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…