All Questions
Tagged with gradient-flows ds.dynamical-systems
12 questions
5
votes
3
answers
643
views
What quantities are conserved under a general gradient-flow $\dot X(t) = -\nabla L(X(t))$?
Let $L:\mathbb R^N \to \mathbb R$ be a continuously differential function with gradient $x \mapsto \nabla L(x)$ and consider induced gradient-flow
$$
\dot X(t) = -\nabla L(X(t)).
$$
Question. Is ...
- 6,853
5
votes
0
answers
140
views
What kinds of gradient-flows on $\mathbb R^d$ preserve the log-concavity of the distribution $\mu_0$ of starting point $x_0$
Let $\mu_0$ be a log-concave distribution on $\mathbb R^d$ and let $f:\mathbb R^d \to \mathbb R$ be $C^2$. Let $x_0$ be sampled uniformly at random from a log-concave distribution $\mu_0$, meaning ...
- 6,853
5
votes
0
answers
281
views
Basin of attraction of gradient flow
Suppose we have a compact Riemannian manifold $(M,g)$, and a Morse function $f : M \rightarrow \mathbb{R}$. Suppose we consider the gradient flow generated by this function, i.e. $$\dot{x_t} = - \...
- 151
4
votes
0
answers
148
views
Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$
In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
- 91
3
votes
0
answers
93
views
Regularity of center manifold
Consider a $C^r$ vector field $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with $r \geq 1$. Let $\bar x$ be a critical point of $f$, that is, $f(\bar x) = 0$.
Suppose that the spectrum of $\mathrm{D}f(\...
- 31
2
votes
1
answer
291
views
When does uniqueness of a stable equilibrium imply it is globally stable?
Given a gradient dynamical system
$$\dot x=-\nabla f(x),$$
my question is:
(1) If there exists only one equilibrium $x^*$ which is stable (if necessary, this can be changed to stable asymptotically ...
- 405
2
votes
0
answers
74
views
Is it likely that gradient flow trajectories of a $G$-invariant function pass through degenerate points?
This question may be posed somewhat vaguely, but I'm interested to actually get an idea of what to expect, so I try to not target it at a specific result.
Assume that $G$ is a compact Lie group, ...
- 139
2
votes
0
answers
65
views
Chain recurrent points of a gradient-like system
Let $X$ be a compact metric space and $f:X\to X
$ homeomorphism. Let $V:X\to \mathbb{R}$ be a Lyapunov function for $(X,f)$ (continuous function such that $(\forall x\notin Fix(f))\ \ V(f(x))<V(x))...
- 141
2
votes
0
answers
83
views
Center-stable manifold theorem on manifold with boundary
I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary.
Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...
- 21
1
vote
1
answer
181
views
Gradient descent under the presence of symmetries
Let $M$ be a Riemannian manifold (I'm happy to assume it is Euclidean space) with a function $f: M \to \mathbb R$ and a group of isometries $G$ acting on $M$ and preserving $f$, i.e., $f(gm) = f(m)$ ...
- 7,746
1
vote
1
answer
169
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Gradient-like dynamical systems
I've tried asking this question on Mathematics site, but I only got an upvote and no answer. I've searched online, tried to find something about this topic, but I haven't found much (and the things I ...
- 141
0
votes
0
answers
223
views
Convergence of ODE solutions almost everywhere to a stable equilibrium point
Theorem: Suppose ${\bf g} :\mathbb{R}^n \mapsto \mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A} \subset \mathbb{R}^n$ such that $\bf g$ is uniformly Lipschitz on $\...
- 1