Stability theory, including global 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 ...
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64 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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332 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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51 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}^{...
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103 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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98 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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150 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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76 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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42 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 ...
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114 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. ...
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138 views

Mumford's vector bundle stability equivalent the notion orbit stability for a G-space?

Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class ...
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19 views

Criteria for existence of stable principal submatrices of a stable matrix?

Let $A$ be an $n\times n$ real matrix. Suppose $A$ is stable, that is, all the eigenvalues of $A$ have strictly negative real part. Question: What are some results about existence of stable ...
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12 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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Can I study a global stability of equilibrium point in a fractional order system?

I have a system of fractional-order differential equations in the sense of  Caputo’s derivatives. For this model, I obtained the equilibrium points and proved that some equilibrium points are locally ...
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41 views

Mathieu equation instability

Let's have Mathieu equation: $$ \tag 1 \frac{d^2y(x)}{dx^2} +(A -2q\times cos(2x))y(x) = 0, \quad q\geqslant 1, \quad A > 2q $$ This case is different from the famous case $q << 1, A> 2*q$,...
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79 views

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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215 views

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?...