Questions tagged [stability]
Stability theory, including global stability (in dynamical systems, where it can notably be used in combination with ds.dynamical-systems)
162 questions
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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 ...
- 111
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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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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 ...
- 1,675
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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
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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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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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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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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 ...
CommunityBot
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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 ...
- 131
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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 $$ \alpha I \succeq W_t \...
- 24.2k
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Semistability in GIT
If I understand correctly, in geometric invariant theory, polystable points can be defined as those which have a closed orbit. Is it true that semistable points can be characterized as those whose ...
- 1,347
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Semi-stability of Ulrich bundle
A vector bundle $E$ on a smooth projective variety $X$ is called Ulrich bundle if it is Arithmetically Cohen-Macaulay , i.e., $H^i(E(t)) = 0 $ for all $t \in Z$ and $0 < i < k$ and with Hilbert ...
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Hrushovski's proof of the Manin-Mumford Conjecture
For my master's thesis, I am studying Hrushovski's model-theoretic proof of the Manin-Mumford Conjecture. Among the references I have used are the following:
Lecture notes 'Model Theory of Difference ...
- 61
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Stable restrictions of sheaves
Let $X$ be a projective variety and $Y$ a subvariety.
If $E$ is a stable sheaf on $X$, then under certain circumstances (e.g. the theorems of Flenner, Mehta-Ramanathan, Bogomolov) the restriction $E|...
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Why study Bogomolov's T-Stability
Bogomolov introduced the notion of $T$-stability. I know that such stability does not sit in the category of canonical metrics on vector bundles. We know that if a vector bundle admits a Hermitian-...
CommunityBot
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Bogomolov–Miyaoka–Yau inequality for minimal varieties with intermediate Kodaira dimension $0<\kappa (X)<\dim X$
From the differential geometric proof of Yau and the algebraic proof of Miyaoka for minimal varieties of general type $\kappa (X)=\dim X$, we know that $$(-1)^nc_1^n(X)\leq (-1)^n\frac{2(n+1)}{n} c_1^...
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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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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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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 ...
- 3,871
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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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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}
...
- 1,073
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Direct sum of two stable bundles of same slope
How to prove that the direct sum of two stable vector bundles of the same slope over a smooth curve is a semistable bundle?
- 7,902
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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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2
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Higgs bundle and stable bundle
Let $(E,\phi)$ be a $G$-Higgs bundle $\phi\in H^{0}(X,ad(E)\otimes D)$ where $D$ is a divisor on X.
I suppose that $(E,\phi)\in \mathcal{M}^{ani}$ the anisotropic locus.
In particuler, this bundle ...
- 61
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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\...
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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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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 ...
- 20.2k
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How do solutions of a PDE depend on parameters?
Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$).
Let $f\in H^{1/2}(\partial\Omega)=H^1(\Omega)/H^...
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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 ...
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Condition Number and CFL Condition in Finite difference Methods [closed]
when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability:
One factor would be the condition number of the approximation operator. The other factor ...
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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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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}(...
- 1,167
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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
$$
A(t) =
\left(
\begin{...
- 1,167
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Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$
This question has been asked here but there is no answer:
https://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y
Consider autonomous ODE $y' = ...
CommunityBot
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Quadratic stability of linear time varying system
(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \...
CommunityBot
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Existense of semi-stable vector bundles on smooth curves in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic and $X$ be a smooth projective curve over $k$ of genus $g \ge 2$. Fix a polarization $L$ on $X$. Does there exist a semi-stable ...
CommunityBot
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Choosing a group action to do GIT of hypersurfaces
When studying GIT stability of hypersurfaces $d$ of $\mathbb P^n$ we look at the Hilbert Scheme $H=\mathbb P^N$ parametrizing homogeneous polynomials $f_d(x_0,\ldots,x_n)$ of degree $d$. There is ...
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Direct limit of coherent sheaves and semi-stability
Let $R$ be a discrete valuation ring, $\{B_i\}_{i\in I}$ be an inductive system of $R$-algebras of finite type and $B$ the direct limit of the inductive system. Let $X$ be a regular, projective scheme,...
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How to derive explicit bound for the solution of following equation?
Let's have equation
$$
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
$$
How to derive explicit upper bound ...
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428
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Nonlinear ODE system: stability
I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...
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Examples of systems with stable equilibria at the boundary of the phase space
Hopfield networks are gradient dynamical systems, used (among other things) to solve combinatorial optimization problems, because stable equilibria are at vertices of the hypercube $[-1,1]^n$. They ...
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Bounded input Bounded output stability for heat equation
This is a cross-post from Computational Science.
I am interested in proving or obtaining a counterexample to the following conjecture.
Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let ...
CommunityBot
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How to analyse the stability of hyperbolic balance laws with diffusion?
Assume we have the following system of balance laws:
$$ U_t+\partial_x F(U,x)=S(U,x)+\partial_x(D(U,x)U_x). $$
Is there any method to analyse the stability of its solution (assume that the solution ...
- 105
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GKS stability of a finite difference scheme
In this paper, I can not reproduce the results obtained equation 62.
I have tried to reproduce it using Wolfram alpha but the results are different.
However, using equation (40) instead of the one ...
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Stability of minimal surfaces
Let $\Gamma$ be a prescribed $n-2$ dimensional set and assume $S \subset R^n$ is a minimal hyper-surface with respect to some smooth metric $g$ on $R^n$, and $\partial S= \Gamma$. Is $S$ is stable ...
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What are the applications of Grillakis Shatah and Strauss paper?
I am studying the following paper.
Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197.
...
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Lyapunov stability, nonlinear system
Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article.
...
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Non-linear state-space model system stability using Lyapunov?
I have a non-linear system modelled in state-space as follow:
$$
\mathbf {\dot x} = \mathbf A(x) \mathbf x
$$
I need to find out if this system is stable, so I was thinking in using the Lyapunov ...
- 1,590
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0
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Prove that origin is globally exponentially stable with Lyapunov Indirect Method
I'm wondering, if we have a nonlinear system governed by
$\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz
how can we show that the origin is globally exponentially stable?...
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