# Tagged Questions

Stability theory, including global stability

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### 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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### 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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### 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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### 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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### 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 ...
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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 ...
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### Difference between Gieseker semistable and slope semistable

Let $X$ be a projective reduced (not necessarily irreducible) curve over an algebraically closed field and $\mathcal{F}$ be a pure coherent sheaf on $X$. Is it true that $\mathcal{F}$ is Gieseker ...
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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 ...
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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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### 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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### 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. ...