Questions tagged [stability]
Stability theory, including global stability (in dynamical systems, where it can notably be used in combination with ds.dynamical-systems)
76
questions with no upvoted or accepted answers
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views
Why study Bogomolov's T-Stability
Bogomolov introduced the notion of $T$-stability. I know that such stability does not sit in the category of canonical metrics on vector bundles. We know that if a vector bundle admits a Hermitian-...
user21574
6
votes
0
answers
372
views
Hrushovski's proof of the Manin-Mumford Conjecture
For my master's thesis, I am studying Hrushovski's model-theoretic proof of the Manin-Mumford Conjecture. Among the references I have used are the following:
Lecture notes 'Model Theory of Difference ...
- 61
6
votes
0
answers
547
views
Bogomolov–Miyaoka–Yau inequality for minimal varieties with intermediate Kodaira dimension $0<\kappa (X)<\dim X$
From the differential geometric proof of Yau and the algebraic proof of Miyaoka for minimal varieties of general type $\kappa (X)=\dim X$, we know that $$(-1)^nc_1^n(X)\leq (-1)^n\frac{2(n+1)}{n} c_1^...
user21574
5
votes
0
answers
94
views
Stability and capacity, error in the book of Adams-Hedberg?
I am struggling to understand the proof from the book of Adams and Hedberg, "Function spaces and potential theory". It seems to me that there is a serious flaw, and moreover, the statement is ...
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5
votes
0
answers
178
views
Projective unitary flat structures of $\mathbb{P}^1$-bundles on Riemann surfaces
Narasimhan and Seshadri proved a rather surprising result about vector bundles on a compact connected complex manifold $X$. That is
Two holomorphic vector bundles arising from unitary representations ...
- 713
5
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0
answers
286
views
Bundles as Extensions and Jump Phenomena
Let $C$ be a Riemann Surface of genus $g \geq 2$. Consider a Vector Bundle of rank $r$ and degree $d$ on $C$. It is often convenient to construct such a Vector Bundle as an extension
\begin{equation}
...
- 1,063
5
votes
0
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83
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How to derive explicit bound for the solution of following equation?
Let's have equation
$$
y''(t) + \left(\frac{3}{16t^{2}} + \frac{a}{t} -\frac{b}{t^{\frac{5}{4}}}cos(2t)\right)y(t) = 0, \quad t \in (1, \infty), \quad a, b > 0
$$
How to derive explicit upper bound ...
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4
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0
answers
188
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Elementary proof Hilbert-Mumford stability criterion for $\operatorname{GL}_n(\mathbb{C})$
In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \...
- 7,200
4
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0
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The logistic elliptic equation
Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...
- 1,330
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255
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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. $$\dot{x_t} = - \...
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4
votes
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KAM stable orbits are smooth
I'm in my final year of my undergraduate studies doing work on modelling the n-body problem numerically and I also have some interest in theoretical guarantees. Now, I've been looking for a theorem ...
- 3,827
4
votes
0
answers
121
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Conservation laws for modified Degasperis-Procesi equation
It is known that the Korteweg-de Vries equation
$$u_{t}+uu_{x}+u_{xxx} = 0,$$
with $u=u(x,t)$ smooth and with period equal to $L$, has important conservation laws, namely,
$$E(u)=\frac{1}{2}\int_{0}^{...
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4
votes
0
answers
601
views
well-posedness of the transport equation
I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
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3
votes
0
answers
48
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Stability of indefinitely damped mechanical system with diagonal stiffness
I'm trying to find conditions for the asymptotic stability of the following linear system,
\begin{equation}
\mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0
\end{equation}
given the ...
3
votes
0
answers
45
views
Which invertible linear maps preserve the set of Hurwitz stable matrices?
Let $V = M_n(\mathbb{R})$ be the set of all $n\times n$ matrices with real elements and $V_{-}$ be a subset of Hurwitz stable matrices, i.e. matrices such that all their eigenvalues have strictly ...
- 1,264
3
votes
0
answers
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Asymptotic behaviors of equilibrium points of a switching SDE with Levy jumps?
Consider the following paper titled: Stochastic regime switching SIR model driven by Lévy noise, authored by Yingjia Guo.
Link: https://www.sciencedirect.com/science/article/pii/S0378437117302145
The ...
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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:
(...
- 11.6k
3
votes
0
answers
62
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Part 2: When does the maximal root of this polynomial have unit magnitude? Prove an inverse linear relation between parameters
Examine the polynomial
$$
x^{\tau+1}-\left(1+m\right)x^{\tau}+mx^{\tau-1}+\left(1-m\right)\alpha=0\,.
$$
with positive parameters $\tau,\alpha$ and $m<1$, and denote $\left|x_{\max}\left(\tau,\...
- 876
3
votes
0
answers
78
views
conditions for asymptotic comparison to hold
I have the following simple dynamical system:
\begin{align}
x_1' &= a - f(x_2)x_1\\
x_2' &= bx_1 - cx_2,
\end{align}
where all parameters and initial conditions are positive. $f(x_2)$ is a ...
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3
votes
0
answers
188
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 ...
- 1,269
3
votes
0
answers
364
views
Fujita decomposition versus Zariski decomposition
Fujita decomposition: Let $\frak \pi : X \to B$ be a fibration of a compact Kahler manifold $\frak X$ over a projective curve $\frak B$ then $\pi_*\left(K_{\frak X/B}\right)=A\oplus B$ where $A$ is ...
user21574
3
votes
0
answers
279
views
Direct limit of coherent sheaves and semi-stability
Let $R$ be a discrete valuation ring, $\{B_i\}_{i\in I}$ be an inductive system of $R$-algebras of finite type and $B$ the direct limit of the inductive system. Let $X$ be a regular, projective scheme,...
- 2,126
3
votes
0
answers
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views
Choosing a group action to do GIT of hypersurfaces
When studying GIT stability of hypersurfaces $d$ of $\mathbb P^n$ we look at the Hilbert Scheme $H=\mathbb P^N$ parametrizing homogeneous polynomials $f_d(x_0,\ldots,x_n)$ of degree $d$. There is ...
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3
votes
0
answers
181
views
Ampleness on the P^1 bundle over Siegel threefold
I am looking at the Shimura variety for $\mathrm{GSp}_4(\mathbb Q)$, with hyperspecial level structure at $p$. Let $X$ denote the special fiber over $\mathbb F_p$. For simplicity, let us pretend ...
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2
votes
0
answers
57
views
Identifying bifurcation
[![enter image description here]] 1]1I am trying to analyze the bifurcation of a 3D continuous model. For a certain range of parameter values, the origin is always an unstable point, whereas the ...
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2
votes
0
answers
56
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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 ...
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2
votes
0
answers
58
views
Rotation number for multicomponent Schrödinger equation
Rotation number for Schrödinger equation of the form
\begin{equation}
-x''(t) +q(t) x(t) = E x(t)
\end{equation}
was defined in R. Johnson J. Moser "The rotation number for almost periodic ...
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2
votes
0
answers
125
views
Two notions of stability
Let $Q$ be a finite quiver (i.e. an oriented graph). A representation of $Q$ is by definition a module over the path algebra of $Q$. More concretely, a representation associates to every vertex $v \in ...
- 373
2
votes
0
answers
144
views
Stability test for LTV systems by differential Lyapunov inequalities
Consider a linear time-varying system:
\begin{equation}
\dot x(t) = A(t) x(t), \tag{$*$}
\end{equation}
where $A(t)$ is a time-varying block matrix defined as
$$
A(t) =
\begin{bmatrix}
0 & I\\
-\...
- 55
2
votes
0
answers
125
views
Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions
I have a system of nonlinear Volterra integral equations of form
$$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$
and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...
- 123
2
votes
0
answers
74
views
Center-stable manifold theorem on manifold with boundary
I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary.
Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...
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2
votes
0
answers
47
views
(1, 2) stability and Hausdorff dimension
Recall that a set $E \subset \mathbb{R}^n$ is called (1,2) stable if it satisfies the following:
$$
W_0^{1,2}(E) = W_0^{1,2}(E^0),
$$
where $E^0$ is the interior of $E$ and $W_0^{1,2}(E)$ is defined ...
- 93
2
votes
1
answer
291
views
Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?
For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it.
Problem definition: Let $f(\xi) \in \...
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2
votes
0
answers
60
views
Strong stability of the wave equation with time depending potential
It is well known that the wave equation with frictional damping
$$\eqalign{
& {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr
...
- 545
2
votes
0
answers
93
views
Stability of dynamical system via Lyapunov
I have a dynamical system which has the following form:
$\dot x=\mathcal F_1(m_1)x+\mathcal F_2(m_2)x$.
My objective is to find the parameters $m_1$ and $m_2$ via LMI (linear matrix inequality) using ...
2
votes
0
answers
63
views
Exponential Convergence Under a Lyapunov-Like Assumption
Consider $V=\mathbb{R}^d\times\mathbb{R}^n$ with coordinate $x^T=[\theta^T,\sigma^T]$. I have an ODE of the form: $\dot{x}=F(x)$, where $F$ is assumed to be sufficiently smooth.
Suppose that there's ...
- 669
2
votes
0
answers
101
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Stochastic stability of "open" continuous-time stochastic systems: reference request
I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but ...
- 1,675
2
votes
0
answers
327
views
Showing positive stability of a matrix constructed from a positive matrix
A is a positive nonsingular matrix. Let $s>\rho(A)$. We want to show that $B\equiv\left(A^{T}A\right)^{-1}\left(sI-A^{T}\right)$ is a positive stable matrix, i.e., all eigenvalues of this matrix ...
- 21
2
votes
0
answers
48
views
How to analyse the stability of hyperbolic balance laws with diffusion?
Assume we have the following system of balance laws:
$$ U_t+\partial_x F(U,x)=S(U,x)+\partial_x(D(U,x)U_x). $$
Is there any method to analyse the stability of its solution (assume that the solution ...
- 105
1
vote
0
answers
24
views
Identifying Saddle-node bifurcation of a 3D system of ODEs
I am trying to understand and prove the results shown in the following article. However, I am stuck at a point where it is stated that saddle-node bifurcation of periodic orbits occurs at ...
- 53
1
vote
1
answer
80
views
Norm bound in simultaneous stability to semidefinite program
In the context of robust control, I remember hearing that the two following problems are equivalent.
Find $P \succ 0$, such that $A P + P A^{\top} \prec 0$ for all $A \in \mathscr{A}$ where $$\...
- 195
1
vote
0
answers
50
views
On Designing Some Optimal Control Problems
In the context of a dynamical systems, some states may not be attainable with scalar controls from $L^1(0,T)$, but they may be reachable with feedback controls.
If we know that the system is null ...
- 55
1
vote
0
answers
52
views
Basin of attraction comparative statics* using local energy functions?
Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose ...
1
vote
0
answers
24
views
Reduced $H_{\infty}$ problem for nD systems
Let $G(z)$ denote the (rational not necessarily square and unstable) transfer function of an nD system, where $z=(z_{2},...,z_{n})$, of a discrete spatial-temporal recurrence Givone-Roesser type ...
- 21
1
vote
0
answers
47
views
Linear programming robustness to input perturbations
I'm running a linear program whose parametrization depends on the output of a neural network. I was wondering if there exist results on how robust linear programs are towards perturbations in their ...
- 11
1
vote
0
answers
77
views
Clarification on the proof of Lyapunov-Razumikhin asymptotic stability theorem for delayed differential equations
this is my first question here, hope I am in the right place :)
Recently I have been looking at the proof of theorem 4.2 on Razumikhin stability for RFDEs in the book by Jack Hale and Lunel Verduyn: ...
- 11
1
vote
0
answers
34
views
$L^p$-continuity for discrete linear causal systems
Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by:
\begin{...
- 41
1
vote
0
answers
153
views
Selecting a suitable Lyapunov function for the following systems?
i) SI MODEL
Consider
\begin{align}
\frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex]
\frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I
\end{align}
Where $N=S+I$ is the total population.
If ...
- 185
1
vote
0
answers
22
views
The uniqueness of some semistable torsion free sheaves on Fano threefold
Let $X$ be a prime Fano threefold of index one and even genus $g\geq 6$, one can show that the moduli space of torsion free semistable sheaves $M(2,1,m_g)$ with $m_g=\left \lceil{\frac{g+2}{2}}\right \...
- 1,942
1
vote
0
answers
179
views
Subset of the domain of attraction
Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$
$$
\frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t))
$$
be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
- 61