Skip to main content

Questions tagged [stability]

Stability theory, including global stability (in dynamical systems, where it can notably be used in combination with ds.dynamical-systems)

Filter by
Sorted by
Tagged with
0 votes
0 answers
13 views

Numerical computation of spectral abscissa of operator

I would like to numerically compute the spectral abscissa of an unbounded linear operator $A$ on a Hilbert space. To give you an idea my operator has the form: $$Af(x,y) = a(y) \partial_x f(x,y) - b(x)...
toaster's user avatar
  • 121
2 votes
0 answers
46 views

Stability on manifold with boundary

Let $(X,\partial X)$ a smooth Kahler manifold with boundary, i.e. the interior of $X$ is Kahler, Donaldson proved that: Given a smooth vector bundle $E$ over $X$ such that $E$ is holomorphic over the ...
TaiatLyu's user avatar
  • 345
1 vote
0 answers
42 views

Identifying Saddle-node bifurcation of a 3D system of ODEs

I am trying to understand and prove the results shown in the following article. However, I am stuck at a point where it is stated that saddle-node bifurcation of periodic orbits occurs at ...
SHR's user avatar
  • 53
0 votes
0 answers
67 views

Proving Hopf bifurcations for 3D system

I am working with a 3D continuous system of ODEs. I have found Hopf bifurcation numerically for a certain value of parameter. However, I want prove it analytically. Is it enough to show that the ...
SHR's user avatar
  • 53
2 votes
0 answers
59 views

Identifying bifurcation

[![enter image description here]] 1]1I am trying to analyze the bifurcation of a 3D continuous model. For a certain range of parameter values, the origin is always an unstable point, whereas the ...
SHR's user avatar
  • 53
2 votes
0 answers
59 views

Where can I find resources for a paper "Stability analysis of a novel DDE of HIV CD4+ T-cells"?

I am currently working on a the paper [NND]: Question: On page 4, equation 6 introduces a concept related to the infection rate within the context of the HIV model. Unfortunately, the paper does not ...
Furdzik Zbignew's user avatar
2 votes
1 answer
93 views

Norm bound in simultaneous stability to semidefinite program

In the context of robust control, I remember hearing that the two following problems are equivalent. Find $P \succ 0$, such that $A P + P A^{\top} \prec 0$ for all $A \in \mathscr{A}$ where $$\...
wsz_fantasy's user avatar
4 votes
0 answers
205 views

Elementary proof Hilbert-Mumford stability criterion for $\operatorname{GL}_n(\mathbb{C})$

In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \...
Libli's user avatar
  • 7,250
3 votes
0 answers
49 views

Stability of indefinitely damped mechanical system with diagonal stiffness

I'm trying to find conditions for the asymptotic stability of the following linear system, \begin{equation} \mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0 \end{equation} given the ...
Shivang Rawat's user avatar
0 votes
0 answers
25 views

Coefficients in the series expansion of a central manifold are all zero

I have a system of 4 ODEs, which linearized around the origin gives $$ \begin{align} &\dot{q_1}=a\, q_1\\ &\dot{q_2}=b\,q_2\\ &\dot{q_3}=0\\ &\dot{q_4}=c\,q_4 \end{align} $$ with $a$, $...
F.Mor's user avatar
  • 1
2 votes
1 answer
37 views

Questions on the "generalized" min. singular value of $A$ given $B$: $\min_{L \in \mathbb{R}^{n \times m}} \{\|BL\|_F: \det(A + BL) = 0\}$

Let $A \in \mathbb{R}^{n \times n}$ be a matrix. Recall $\sigma_{\min}(A)$ is the Frobenius distance between $A$ and the set of singular matrices: $$\sigma_\min(A) = \min_{E \in \mathbb{R}^{n \times n}...
Spencer Kraisler's user avatar
3 votes
0 answers
46 views

Which invertible linear maps preserve the set of Hurwitz stable matrices?

Let $V = M_n(\mathbb{R})$ be the set of all $n\times n$ matrices with real elements and $V_{-}$ be a subset of Hurwitz stable matrices, i.e. matrices such that all their eigenvalues have strictly ...
Fiktor's user avatar
  • 1,284
3 votes
2 answers
285 views

Stability results for general linear stochastic ODE

I am interested in the following time-invariant multivariate SDE: \begin{equation} dx_i = \sum_{j} a_{ij} x_j\,dt + \sum_{j,k} b_{ijk} x_k \, dW_j \end{equation} Despite its simplicity the general ...
Panopticon's user avatar
2 votes
0 answers
59 views

Rotation number for multicomponent Schrödinger equation

Rotation number for Schrödinger equation of the form \begin{equation} -x''(t) +q(t) x(t) = E x(t) \end{equation} was defined in R. Johnson J. Moser "The rotation number for almost periodic ...
0x11111's user avatar
  • 493
1 vote
0 answers
51 views

On Designing Some Optimal Control Problems

In the context of a dynamical systems, some states may not be attainable with scalar controls from $L^1(0,T)$, but they may be reachable with feedback controls. If we know that the system is null ...
elmas's user avatar
  • 55
2 votes
0 answers
135 views

Two notions of stability

Let $Q$ be a finite quiver (i.e. an oriented graph). A representation of $Q$ is by definition a module over the path algebra of $Q$. More concretely, a representation associates to every vertex $v \in ...
Laurent Cote's user avatar
2 votes
0 answers
316 views

open problem in numerical analysis [closed]

I am interested in open and current issues in numerical analysis, there are good references in this respect. Thanks for your response
Lahcen El-ouadefli's user avatar
3 votes
1 answer
233 views

Stability analysis of equilibrium point of non-linear ODE system with Jacobian going to infinity

Let's say I have a nonlinear system of ODEs, where one of equations looks like: $$ \frac{dX_i}{dt} = a_1X_0+\dotsb+a_j\frac{X_j^{0.5}}{X_j^{0.5}+b_j^{0.5}}+\dotsb. $$ And equilibrium point is 0. I ...
Omega's user avatar
  • 31
3 votes
2 answers
185 views

Floquet coefficients under time change

Let's consider two ODEs $\tag{1}\label{1}\frac{du}{dt}=\gamma(u(t))\ F(u(t))$ and $\tag{2}\label{2}\frac{dv}{d\tau}=F(v(\tau))$ where $f\in C^\infty(\mathbb R^n,\mathbb R^n)$ and $\gamma\in C^\infty(\...
herve's user avatar
  • 41
3 votes
1 answer
157 views

Does gravity constant affect boundedness of solution?

Consider a second order gradient-like system with linear damping $$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$ Suppose $f\in C^2(\mathbb{R}^n)$ and $\inf_{x\in\mathbb{R}^n}f(x)&...
Jean Legall's user avatar
0 votes
0 answers
63 views

Modification of a lemma on the boundness of a stochastic process

Lemma 1 is widely used in the stability proof of stochastic process. Lemma 1 Assume that $\xi_k$ is a stochastic process and there is a stochastic process $V(\xi_k)$ as well as real numbers $\upsilon_{...
themarshal's user avatar
1 vote
1 answer
218 views

Some question about (semi-)stable sheaves

Let $X$ be a projective normal variety over $\mathbb C$, I have several questions about semi-stable sheaves: Question 1. Suppose that $E$ is a pure sheaf such that $HN_*(E)$ is the Harder-Narasimhan ...
Li Yutong's user avatar
  • 3,392
3 votes
0 answers
133 views

Asymptotic behaviors of equilibrium points of a switching SDE with Levy jumps?

Consider the following paper titled: Stochastic regime switching SIR model driven by Lévy noise, authored by Yingjia Guo. Link: https://www.sciencedirect.com/science/article/pii/S0378437117302145 The ...
Math's user avatar
  • 185
4 votes
0 answers
105 views

The logistic elliptic equation

Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form: $$ \begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...
Bogdan's user avatar
  • 1,506
5 votes
1 answer
271 views

Unbounded solution but bounded Euler discretization

Is there an ordinary differential equation in $\mathbb{R}^d$ induced by a gradient vector field with unbounded solutions, for which the difference equations obtained by using the forward Euler method ...
Jean Legall's user avatar
1 vote
0 answers
52 views

Basin of attraction comparative statics* using local energy functions?

Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose ...
cluelessmathematician's user avatar
1 vote
1 answer
123 views

Analytically characterizing basins of attraction boundaries and sizes

While I understand that doing the above is not possible in general, I would like to know more about how to proceed when it is possible. That is, what are the common methods people use to analytically ...
cluelessmathematician's user avatar
1 vote
0 answers
24 views

Reduced $H_{\infty}$ problem for nD systems

Let $G(z)$ denote the (rational not necessarily square and unstable) transfer function of an nD system, where $z=(z_{2},...,z_{n})$, of a discrete spatial-temporal recurrence Givone-Roesser type ...
Yossi Peretz's user avatar
1 vote
0 answers
51 views

Linear programming robustness to input perturbations

I'm running a linear program whose parametrization depends on the output of a neural network. I was wondering if there exist results on how robust linear programs are towards perturbations in their ...
f.k's user avatar
  • 11
1 vote
0 answers
85 views

Clarification on the proof of Lyapunov-Razumikhin asymptotic stability theorem for delayed differential equations

this is my first question here, hope I am in the right place :) Recently I have been looking at the proof of theorem 4.2 on Razumikhin stability for RFDEs in the book by Jack Hale and Lunel Verduyn: ...
Adriano Mele's user avatar
3 votes
1 answer
150 views

Stable periodic orbits for three equal masses

For three equal masses in any number of dimensions (this might not be important, but 2D or 3D or 4D is fine) under just classical gravity (i.e., inverse-square force law), what stable periodic orbits ...
bobuhito's user avatar
  • 1,547
10 votes
1 answer
417 views

Symmetric polynomials that detect positivity

Imagine there are numbers $a_1,\ldots,a_n \in \mathbb R$ and you want to know whether they are all positive. You cannot access the numbers themselves, but you can choose any symmetric polynomials you ...
Louis Deaett's user avatar
  • 1,513
1 vote
1 answer
135 views

Stability of certain second order ODE

I am having a hard time determining the motion of $X$ in the ODE $X''+\nabla f(X)=0$ with initial conditions arbitrary $X(0)$ and zero velocity, i.e. $X'(0) = 0$. $X$ is in $\mathbb{R}^{n}$, and $f$ ...
MMH..'s user avatar
  • 11
3 votes
1 answer
461 views

Why is the largest invariant set the following?

Consider this paper: Hai-Feng Huo and Li-Xiang Feng, "Global stability for an HIV/AIDS epidemic model with different latent stages and treatment", Applied Mathematical Modelling, Volume 37, ...
Math's user avatar
  • 185
1 vote
1 answer
329 views

Why should we model infectious diseases with fractional differential equations?

With COVID19 becoming a pandemic I saw some researchers trying to model it with fractional differential equations (FDE) instead of ordinary differential equations (ODE). From a technical standpoint I ...
SetofMeasureZero's user avatar
2 votes
0 answers
146 views

Stability test for LTV systems by differential Lyapunov inequalities

Consider a linear time-varying system: \begin{equation} \dot x(t) = A(t) x(t), \tag{$*$} \end{equation} where $A(t)$ is a time-varying block matrix defined as $$ A(t) = \begin{bmatrix} 0 & I\\ -\...
Evan's user avatar
  • 55
0 votes
1 answer
312 views

Conditions for a block matrix to be Hurwitz stable

Consider the following block matrix: $$ A = \begin{bmatrix} 0 & I\\ -M & -I \end{bmatrix} $$ Suppose matrix $M$ is positive definite and satisfies $M\succeq \alpha I$, where $\alpha>0$ is a ...
Evan's user avatar
  • 55
1 vote
0 answers
34 views

$L^p$-continuity for discrete linear causal systems

Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by: \begin{...
avril_14th's user avatar
1 vote
0 answers
154 views

Selecting a suitable Lyapunov function for the following systems?

i) SI MODEL Consider \begin{align} \frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex] \frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I \end{align} Where $N=S+I$ is the total population. If ...
Math's user avatar
  • 185
1 vote
0 answers
23 views

The uniqueness of some semistable torsion free sheaves on Fano threefold

Let $X$ be a prime Fano threefold of index one and even genus $g\geq 6$, one can show that the moduli space of torsion free semistable sheaves $M(2,1,m_g)$ with $m_g=\left \lceil{\frac{g+2}{2}}\right \...
user41650's user avatar
  • 1,962
2 votes
2 answers
1k views

Routh-Hurwitz criterion for matrices

The Routh-Hurwitz criterion explicitly specifies a finite set of inequalities on the coefficients of a polynomial, necessary and sufficient that all zeros lie in the unit circle or in the left half ...
Arnold Neumaier's user avatar
4 votes
1 answer
90 views

Do Pareto critical points of a multicriteria optimization problem form an attractor of the dynamical system induced by a descent algorithm?

Let $d\in\mathbb N$, $k\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Say that $v\in\mathbb R^d$ is a descent direction at $x\in\mathbb R^d$ if ${\rm D}f(x)v<0$ (component-wise) ...
0xbadf00d's user avatar
  • 157
1 vote
0 answers
181 views

Subset of the domain of attraction

Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$ $$ \frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t)) $$ be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
SampleTime's user avatar
2 votes
0 answers
125 views

Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions

I have a system of nonlinear Volterra integral equations of form $$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$ and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...
Quiet_waters's user avatar
0 votes
0 answers
107 views

Reference for matrix Lyapunov function / matrix dynamic system / stability

We usually consider $\dot{x} = f(x)$, where $x$ is a vector. Now, I want to consider $$\dot{X}=f(X,U),$$ where $X$ is a square matrix $\mathbb{R}^{n\times n}$ state, $U$ is a square matrix variable $\...
Denny's user avatar
  • 101
1 vote
0 answers
61 views

Lyapunov theory in coupled nonlinear dynamic system with input

Suppose I have the following nonlinear coupled dynamic system \begin{align*} &\dot{x}_1 = f_1(x_1,x_2)\\ &\dot{x}_2 = f_2(x_2) + u \end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{...
sleeve chen's user avatar
3 votes
0 answers
65 views

Uniform stability of linear operators - reference request

Let $T$ be a bounded linear operator on a complex Banach space $X$. I am looking for a reference for the following result: Theorem 1. Let $p \in [1,\infty]$. The following assertions are equivalent: (...
Jochen Glueck's user avatar
1 vote
0 answers
44 views

Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system

Consider the initial value problem \begin{equation}\label{fainait ve} \dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; t \geq 0, \; \;\boldsymbol{f}(\boldsymbol{0}_n) = \boldsymbol{0}...
Johannes's user avatar
1 vote
0 answers
122 views

Von Neumann analysis on a finite difference hyperbolic scheme

I am doing a Von Neumann analysis on a staggered finite difference scheme (for Maxwell's Equations). The finite difference scheme is: $$ \mathbf{u}_v|^{n+2}_{i,j} - \mathbf{u}_v|^{n}_{i,j} = - A \frac{...
Barros's user avatar
  • 11
0 votes
0 answers
38 views

Terminology: Almost stable states

I have a question about fixed points which are almost stable. I have an increasing transition function $f:[0,1]\rightarrow[0,1]$ where $f(0)>0$ and $f(1)<1$ but I don't necessarily have ...
MDR's user avatar
  • 188