Related papers: Generic Singularities of 3D Piecewise Smooth Dynam…
Roughly speaking, the problem of geography asks for the existence of varieties of general type after we fix some invariants. In dimension $1$, where we fix the genus, the geography question is trivial, but already in dimension $2$, it…
We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena…
In [1], the authors have studied stability of certain causal properties of space-times in general relativity. As a continuation of this work, in the present paper, we review and discuss, some more aspects of stability which occur in various…
We fully generalize a previously-developed computational geometry tool [1] to perform large-scale simulations of arbitrary two-dimensional faceted surfaces $z = h(x,y)$. Our method uses a three-component facet/edge/junction storage model,…
This is a survey on the local structure about a fixed point of discrete finite-dimensional holomorphic dynamical systems, discussing in particular the existence of local topological conjugacies to normal forms, and the structure of local…
The behaviour of a solid-liquid-gas system near the three-phase contact line is considered using a diffuse-interface model with no-slip at the solid and where the fluid phase is specified by a continuous density field. Relaxation of the…
Topological invariants have proved useful for analyzing emergent function as they characterize a property of the entire system, and are insensitive to local details, disorder, and noise. They support boundary states, which reduce the system…
We develop a general compactification framework to facilitate analysis of nonlinear nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded…
Slow-fast dynamical systems, i.e., singularly or non-singularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating…
We prove that certain asymptotically flat initial data sets with nontrivial topology and/or differentiable structure collapse to form singularities. The class of such initial data sets is characterized by a new smooth invariant, the maximal…
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for…
This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
A theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started since a…
Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in…
The necessity and benefit of singular solutions in the study of physical systems is shown. By singular solutions we mean solutions that are not contained in the general solution of the system of equations that describes the dynamic system…
A nonsmooth fold is where an equilibrium or limit cycle of a nonsmooth dynamical system hits a switching manifold and collides and annihilates with another solution of the same type. We show that beyond the bifurcation the leading-order…
Work on standard piecewise-smooth (PWS) dynamical systems, with codimension-1 discontinuity sets, relies on the Filippov framework, which does not always readily generalise to systems with higher codimension discontinuities. These higher…
We introduce a notion of autonomous dynamical systems and apply it to prove rigidity of partially hyperbolic diffeomorphisms on closed compact three-manifolds under some smoothness hypothesis of their associated framing.
The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $\epsilon\rightarrow…