Related papers: Spectral analysis of morse-smale gradient flows
In our paper we construct a new infinite family of symmetries of the Whitham equations (averaged Korteveg-de-Vries equation). In contrast with the ordinary hydrodynamic-type flows these symmetries are nonhomogeneous (i.e. they act…
We investigate the broadband turbulent dynamics of attached and separated flows over a Gaussian bump, focusing on the origin of low-frequency coherent structures. The analysis combines time-resolved experimental measurements with…
This report is actually the in-class project of MATH489 FA2022, Dynamics and Differential Equations in University of Illinois Urbana-Champaign. The report is mainly about gradient flow of a Morse-type functions (Morse functions, Morse--Bott…
Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and co-index. For any critical point x choose an integer a(x) arbitrarily. Then there exists a…
In this paper we develop a Morse theory for the uniform energy. We use the one-sided directional derivative of the distance function to study the minimizing properties of variations through closed geodesics. This derivative is then used to…
We develop a geometrical micro-local analysis of contact Anosov flow, such as geodesic flow on negatively curved manifold. This micro-local analysis is based on wave-packet transform discussed in arXiv:1706.09307. The main result is that…
The recent introduction of the gradient flow has provided a new tool to probe the dynamics of quantum field theories. The latest developments have shown how to use the gradient flow for the exploration of symmetries, and the definition of…
The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $\mu$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient…
We consider a vector field $X$ on a closed manifold which admits a Lyapunov one form. We assume $X$ has Morse type zeros, satisfies the Morse--Smale transversality condition and has non-degenerate closed trajectories only. For a closed one…
Understanding feature-outcome associations in high-dimensional data remains challenging when relationships vary across subpopulations, yet standard methods assuming global associations miss context-dependent patterns, reducing statistical…
The complete invariant for gradient like Morse-Smale dynamical systems (vector fields and diffeomorphisms) on closed 4-manifolds are constructed. It is same as Kirby diagram in a case of polar vector field without fixed points of index 3.
Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex $C$, from which topological and geometrical informations…
We show that the classifying space of the flow category of a \emph{tame} Morse function on a smooth, closed manifold $M$ recovers the homotopy type of $M$, thereby addressing a claim in a preprint of Cohen--Jones--Segal. The tameness…
The Teichm\"uller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to…
In this paper, we develop methods for calculating equivariant homology from equivariant Morse functions on a closed manifold with the action of a finite group. We show how to alter $G$-equivariant Morse functions to a stable one, where the…
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $\Phi: \mathbb{R}\times X\to X$ on pfaffian set $X$ is tame if the graph of $\Phi$ is a pfaffian subset of…
In this paper we first construct smooth Morse-Lyapunov functions of attractors for nonsmooth dynamical systems. Then we prove that all open attractor neighborhoods of an attractor have the same homotopy type. Based on this basic fact we…
We investigate solutions of the 2d incompressible Euler equations, linearized around steady states which are radially decreasing vortices. Our main goal is to understand the smoothness of what we call the spectral density function…
We give a transparent derivation of a relation obtained using a supersymmetric non-linear sigma model by Andreev and Altshuler [Phys. Rev. Lett. 72, 902, (1995)], which connects smooth and oscillatory components of spectral correlation…
In this paper we present a new approach to Morse theory based on the de Rham-Federer theory of currents. The full classical theory is derived in a transparent way. The methods carry over uniformly to the equivariant and the holomorphic…