# Questions tagged [stability]

Stability theory, including global stability

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### Stability of the Langevin semigroup under $C_c^\infty(\mathbb R)$

Let
$h\in C^2(\mathbb R)$
$(X^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$ be a continuous process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$X^x_t=x-\frac12\int_0^th'(X^x_s)...

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77 views

### Existence of a Lyapunov function for $-h'\varphi'+\varphi''$ where $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz

Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ The formal adjoint of $L$ is $$L^\ast\psi:=\psi''+(h'...

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**1**answer

45 views

### Marginal stability of discrete linear time-invariant system

I have a question about marginal stability of a system:
\begin{equation}
\mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1]
\end{equation}
I would adapt the definition of marginal stability from this ...

**3**

votes

**1**answer

138 views

### Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?

Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...

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67 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 ...

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**1**answer

165 views

### Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question.
In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator?
Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space.
...

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56 views

### von Neumann stability vs eigenvalues of amplification matrix

My, limited, understanding of the stability analysis of PDEs is that broadly speaking there are two methods: von Neumann analysis which looks at the growth the error of the solution, as described here
...

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44 views

### Center Manifold Theorem and case of all zero eigenvalue

Is the center manifold theorem applicable if say for a planar(2D) system of non-linear ode, the stability matrix has both eigenvalues zero? Of course, there is only one eigenvector.
If not, what is ...

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**1**answer

610 views

### (In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations
$$
\dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t) = -\gamma x_2(t) - \cos(\...

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63 views

### Boundary controllability of the heat equation and observation

I'm studying the boundary controllability of the heat equation
\begin{array}{c}
y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\
y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\
u(...

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**1**answer

56 views

### Stability of eigenvectors for diagonal perturbations

In a previous question I asked about the stability of eigenvalues with respect to diagonal perturbations. Following results from the book Matrix Analysis (by Roger A. Horn & Charles R. Johnson) ...

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62 views

### Showing a modified system of quadratic equations is stable

I have and $n$ dimensional dynamical system, given by
$\dot{x} = M D(x) P x - \frac{c}{2}x$
$P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...

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**1**answer

111 views

### On local attractivity of a coupled non-linear differential equation

Consider a dynamical system described by the following coupled non-linear differential equation
\begin{align}
\dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\
\dot{x}_2(...

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106 views

### Stability when linearization fails

The dynamics of the $j$th system:
\begin{equation}
\begin{split}
\dot{\overline r}_j &= h (\overline r_j)
\,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...

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**1**answer

247 views

### Almost sure stability of a scalar, nonautonomous, nonlinear SDE

I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it.
Maybe there is someone here who can either close the gap in one of the existing answers or has ...

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119 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 ...

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55 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 ...

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**4**answers

4k views

### Is there a mathematical explanation for this cube packing phenomenon?

I saw this unintuitive result on dice packing:
A jumble of thousands of cubic dice, agitated by an oscillating
rotation, can rapidly become completely ordered, a result that is hard
to produce ...

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**0**answers

74 views

### 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 ...

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66 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 ...

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**1**answer

277 views

### Relation between controllability and stability of PDE

In general, when we talk about controllability, we talk about proving the existence of a control input that transfers the state to a desired state at a desired time $T$. However, when we talk about ...

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51 views

### Ratio dependent predator prey model

In the article on Global qualitative analysis of a ratio-dependent predator–prey system- Kuang, 1998
The system is
where a, K, c, m, f, d
are positive constants that stand for prey intrinsic ...

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votes

**1**answer

63 views

### Boundedness of particle motion with time-varying force

Consider the differential equation
$$ m \ddot{x} + k \dot{x} = - W_t x $$
where
$m$ and $k$ are nonnegative.
$x_t \in \mathbb{R}^n$
$W_t$ is a matrix that satisfies $$ \alpha I \succeq W_t \...

**3**

votes

**1**answer

195 views

### Semi-stability of Ulrich bundle

A vector bundle $E$ on a smooth projective variety $X$ is called Ulrich bundle if it is Arithmetically Cohen-Macaulay , i.e., $H^i(E(t)) = 0 $ for all $t \in Z$ and $0 < i < k$ and with Hilbert ...

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267 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 ...

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84 views

### Stable restrictions of sheaves

Let $X$ be a projective variety and $Y$ a subvariety.
If $E$ is a stable sheaf on $X$, then under certain circumstances (e.g. the theorems of Flenner, Mehta-Ramanathan, Bogomolov) the restriction $E|...

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291 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-...

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449 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^...

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250 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 ...

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113 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} = - \...

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**1**answer

92 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 $\...

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118 views

### 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 ...

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113 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.
...

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votes

**2**answers

517 views

### Properties of matrix exponential without using Jordan normal forms

There are some equivalent statements in the classical stability theory of linear homogeneous differential equations $ \dot{x} = Ax, x \in \mathbb{R}^n $ such as:
All eigenvalues of $A$ have negative ...

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193 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}
...

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189 views

### Maximizing the ratio of largest eigenvalues

Let $K$ be a real stable matrix; more specifically,
$$
K=\left(\begin{array}{rrrrr}
0&1&0&\ldots&0\\
0&0&1&\ldots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\...

**4**

votes

**2**answers

272 views

### Conditions for convergence to non-isolated fixed points

Consider a dynamical system of the form
$$
\dot{x}=f(x), \quad x\in X,
$$
and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov $V(x)$ ...

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101 views

### 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}^{...

**4**

votes

**1**answer

225 views

### Decay estimates for wave and Klein-Gordon equation in “generic” curved backgrounds

Consider a $d$-dimensional smooth Lorentzian manifold $(M,g)$ (we assume $d\geq 3$, $M$ to be Hausdorff, paracompact and connected, hence second-countable, and that the signature convention of $g$ is $...

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192 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 ...

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**1**answer

86 views

### One problem about tower stability [closed]

Some years ago i asked myself a question that I still can not answer. Here it is:
A given tower consists of finite homogeneous cubic blocks staying one on another and equal to each other. What is ...

**0**

votes

**1**answer

107 views

### Linearized stream function

I am trying to work through a paper Instability in Parallel Flows Revisited by Friedlander and Howard, and there are a couple steps in the beginning that I do not understand. I apologize in advance ...

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**1**answer

78 views

### Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$

This question has been asked here but there is no answer:
https://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y
Consider autonomous ODE $y' = ...

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**1**answer

263 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 ...

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168 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,...

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152 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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77 views

### 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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votes

**1**answer

90 views

### A Statement from Brauer and Nohel's book on stability of time-depending linear systems

On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a $2 \times 2$ counterexample by Vinograd to the system $y'=A(t)y$, where
$$
A(t) =
\left(
\begin{...

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votes

**2**answers

239 views

### Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...

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votes

**1**answer

323 views

### Show that 0 is Lyapunov stable by using the given Hamiltonian $H(z)$ as a Lyapunov-function

Good day,
This is my first question, I hope all information is given. If not, feel free to ask. Currently I am reading the paper "Stability of relative equilibria in the problem of N+1 vortices" by ...