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5 votes
1 answer
211 views

Stability of ODEs with polynomial nonlinearity

Consider the following ODE system: $$ x′=f(x)\iff \begin{pmatrix} x_1^\prime \\ \vdots\\ x_k^\prime\\ \vdots\\ x_n ^\prime \end{pmatrix} = \begin{pmatrix} f_1(x) \\ \vdots\\ f_k(x)\\ \vdots\\ f_n(x) \...
Zhang Yuhan's user avatar
5 votes
1 answer
507 views

Existense of semi-stable vector bundles on smooth curves in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic and $X$ be a smooth projective curve over $k$ of genus $g \ge 2$. Fix a polarization $L$ on $X$. Does there exist a semi-stable ...
Ron's user avatar
  • 2,126
4 votes
1 answer
118 views

almost linear ODE

Let $A,B$ be $n\times n$ matrices. I am interested in the following ODE in $\mathbb{R}^n$ $$ \frac{dx_t}{dt}=Ax_t+Bx^+_t $$ where $x_t^+=(x^+_{1,t},...,x^+_{n,t})$ and $(\cdot)^+$ is the rectifier: $...
Conformal's user avatar
  • 315
3 votes
1 answer
202 views

Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations

Consider the following autonomous system of differential equations: $$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$ where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
valle's user avatar
  • 884
3 votes
1 answer
278 views

Lyapunov stability of linear system

Consider a linear ODE system $$\dot x_k=\sum_{j=1}^ma_{kj}(t)x_j,\qquad k=1,\ldots, m,\quad a_{kj}(t)\in C[0,\infty).\quad (1)$$ Proposition. Suppose that $$\sup_{t\ge 0}\Big\{\int_0^t\Big(a_{kk}(...
Clive's user avatar
  • 31
3 votes
0 answers
65 views

Uniform stability of linear operators - reference request

Let $T$ be a bounded linear operator on a complex Banach space $X$. I am looking for a reference for the following result: Theorem 1. Let $p \in [1,\infty]$. The following assertions are equivalent: (...
Jochen Glueck's user avatar
3 votes
0 answers
198 views

Asymptotic stability of eigenvalues by compact perturbations

I need some references concerning the asymptotic stability of eigenvalues by compact perturbations. In [T. Kato, Perturbation theory for linear operators] there are some results concerning stability ...
Appliqué's user avatar
  • 1,329
2 votes
0 answers
56 views

Stability on manifold with boundary

Let $(X,\partial X)$ a smooth Kahler manifold with boundary, i.e. the interior of $X$ is Kahler, Donaldson proved that: Given a smooth vector bundle $E$ over $X$ such that $E$ is holomorphic over the ...
TaiatLyu's user avatar
  • 395
2 votes
0 answers
65 views

Where can I find resources for a paper "Stability analysis of a novel DDE of HIV CD4+ T-cells"?

I am currently working on a the paper [NND]: Question: On page 4, equation 6 introduces a concept related to the infection rate within the context of the HIV model. Unfortunately, the paper does not ...
Furdzik Zbignew's user avatar
2 votes
0 answers
373 views

open problem in numerical analysis [closed]

I am interested in open and current issues in numerical analysis, there are good references in this respect. Thanks for your response
Lahcen El-ouadefli's user avatar
1 vote
0 answers
52 views

Stability of Euler discretization

I am looking at the discretization of an ODE: $$x_{n+1} = x_n + \alpha f(x_n),$$ where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...
N. Gast's user avatar
  • 562
1 vote
1 answer
151 views

Analytically characterizing basins of attraction boundaries and sizes

While I understand that doing the above is not possible in general, I would like to know more about how to proceed when it is possible. That is, what are the common methods people use to analytically ...
cluelessmathematician's user avatar
1 vote
0 answers
154 views

Lyapunov stability for nonlinear PDEs

Where can I find a theorem about Lyapunov stability for the equation in Hilbert space? $$ y' = Fy, $$ where $F : V \to V'$ is a nonlinear operator , $y' \in L^2(0,T,V')$, $V$ is a Hilbert space. ...
jokersobak's user avatar
1 vote
0 answers
176 views

Lyapunov stability, nonlinear system

Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article. ...
freddy's user avatar
  • 21