Tagged Questions

Stability theory, including global stability

100 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 ...
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 ...
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 ...
51 views

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, $$E(u)=\frac{1}{2}\int_{0}^{... 0answers 103 views Direct limit of coherent sheaves and semi-stability Let R be a discrete valuation ring, \{B_i\}_{i\in I} be an inductive system of R-algebras of finite type and B the direct limit of the inductive system. Let X be a regular, projective scheme,... 0answers 98 views Choosing a group action to do GIT of hypersurfaces When studying GIT stability of hypersurfaces d of \mathbb P^n we look at the Hilbert Scheme H=\mathbb P^N parametrizing homogeneous polynomials f_d(x_0,\ldots,x_n) of degree d. There is ... 0answers 150 views Ampleness on the P^1 bundle over Siegel threefold I am looking at the Shimura variety for \mathrm{GSp}_4(\mathbb Q), with hyperspecial level structure at p. Let X denote the special fiber over \mathbb F_p. For simplicity, let us pretend ... 0answers 76 views Showing positive stability of a matrix constructed from a positive matrix A is a positive nonsingular matrix. Let s>\rho(A). We want to show that B\equiv\left(A^{T}A\right)^{-1}\left(sI-A^{T}\right) is a positive stable matrix, i.e., all eigenvalues of this matrix ... 0answers 42 views How to analyse the stability of hyperbolic balance laws with diffusion? Assume we have the following system of balance laws:$$ U_t+\partial_x F(U,x)=S(U,x)+\partial_x(D(U,x)U_x). $$Is there any method to analyse the stability of its solution (assume that the solution ... 0answers 114 views Lyapunov stability, nonlinear system Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article. ... 0answers 138 views Mumford's vector bundle stability equivalent the notion orbit stability for a G-space? Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class ... 0answers 19 views Criteria for existence of stable principal submatrices of a stable matrix? Let A be an n\times n real matrix. Suppose A is stable, that is, all the eigenvalues of A have strictly negative real part. Question: What are some results about existence of stable ... 0answers 12 views Conditions for convergence to non-isolated fixed points Consider a dynamical system of the form$$ \dot{x}=f(x), \quad x\in X, $$and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov V(x) ... 0answers 11 views Can I study a global stability of equilibrium point in a fractional order system? I have a system of fractional-order differential equations in the sense of Caputo’s derivatives. For this model, I obtained the equilibrium points and proved that some equilibrium points are locally ... 0answers 41 views Mathieu equation instability Let's have Mathieu equation:$$ \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 $q << 1, A> 2*q$,...
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?...