Questions tagged [stability]

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

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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 ...
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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: ...
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3 votes
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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 ...
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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 ...
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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 $f$ ...
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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, ...
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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 ...
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2 votes
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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\\ -\...
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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 ...
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$L^p$-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{...
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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 ...
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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 \...
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Perron - like theorem for LTV systems & BIBO stable

I'm looking for a necessary and sufficient conditions such that a LTV control system $\dot{x}(t) = A(t) x(t) + u(t), \ x(0)=x_0$, for all $t\geq 0$, $y(t) = C(t) x(t)$ satisfies the Perron-like result ...
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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 ...
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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) ...
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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 ...
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2 votes
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115 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 ...
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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 $\...
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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{...
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3 votes
0 answers
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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: (...
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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)), \;\; \boldsymbol{x}(t) \in \mathbb{R}^n, \;\; t \geq 0, \; \...
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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{...
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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 ...
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1 answer
109 views

Observability inequality for the 1D transport equation

Let $(a,b) \subset (0,1)$. Consider the following transport equation $$z_t+z_x=0, \ (t,x)\in(0,T)\times(0,1), \\z(t,0)=0, \ z(0,x)=z_0(x).$$ It is clear that the solution to the above equation is ...
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3 votes
1 answer
162 views

Exact solution to a periodic linear ODE sought

We have been studying a Hamiltonian system that possesses a one-parameter family of periodic orbits, depending on the energy level $h$. We "know" via various non-rigorous means that these ...
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2 votes
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Center-stable manifold theorem on manifold with boundary

I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary. Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...
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Stability of a continuous piecewise linear map

I am studying random perturbation of a system that is continuous and piecewise linear. More precisely: I am given a map $\Phi_1:\mathbb{R}^d\to \mathbb{R}^d$ such that $$ \Phi_1(x) = \left\{\begin{...
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Stability and capacity, error in the book of Adams-Hedberg?

I am struggling to understand the proof from the book of Adams and Hedberg, "Function spaces and potential theory". It seems to me that there is a serious flaw, and moreover, the statement is ...
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1 vote
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Lower bounds of kappa class functions

I saw in the paper "Smooth Satabilization Implies Coprime Factorization" of Eduardo Sontag the following argument: Given a smooth map $a:\mathbb{R} ^{n}\rightarrow\mathbb{R}^{+}$, let $\rho$ be any ...
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2 votes
1 answer
205 views

Seeking a Lyapunov function for a SIR model with immunity loss

We add the immunity loss to the SIR model and obtain the following autonomous system. $$ \begin{align} s' &= -is+\alpha r \\ i' &= i s - \gamma i\\ r' &= \gamma i-\alpha r \end{align} \...
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2 votes
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(1, 2) stability and Hausdorff dimension

Recall that a set $E \subset \mathbb{R}^n$ is called (1,2) stable if it satisfies the following: $$ W_0^{1,2}(E) = W_0^{1,2}(E^0), $$ where $E^0$ is the interior of $E$ and $W_0^{1,2}(E)$ is defined ...
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1 vote
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What is a suitable state feedback adaptive law in discrete time?

I am trying to find a discrete adaptive law and prove stability in an analogous way to a well-known case in continuous time. Let me give a rough explanation of the background. In continuous time, let ...
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Is there any equivalent Foster-Lyapunov Theorem where expected Lyapunov drift is negative though not uniformly upper bounded by a negative number

I have a process $\{X_t\}_t$, where $x_t$ in $[0,1]$, and a Lyapunov function $V(x)=x$ such that the drift $$E[X_{t+1}|x_t] - x_t < -k(1-x_t)$$ for $x_t \in (1-\epsilon,1]$, where $k>0$. Thus ...
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2 votes
1 answer
96 views

Is there any foster-Lyapunov criterion for time varying Markov processes?

Suppose I have 2 Markov processes with transition kernels Q_1(y|x) and Q_2(y|x). Suppose i also have Lyapunov functions V_1, V_2 for these processes w.r.t. a common set, i.e. there exists a compact ...
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1 vote
1 answer
293 views

Stability of holomorphic vector bundles

I'm reading the book "R. O. Wells Jr. - Differential Analysis on Complex Manifolds" and on page 247 the author claims two things about the stability of holomorphic vector bundles that I'm struggling ...
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1 answer
194 views

Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?

For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it. Problem definition: Let $f(\xi) \in \...
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Attractivity of a system with state-dependent transitions

Let $A\in\mathbb{R}^{n\times n}$ and consider the following dynamical system: $$ \frac{\mathrm{d}x(t)}{\mathrm{d}t} = -x(t)+\max\{0,Ax(t)\}, \ \ \ \ x(0)\in\mathbb{R}^n, $$ where $\max\{\cdot\}$ acts ...
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4 votes
1 answer
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almost linear ODE

Let $A,B$ be $n\times n$ matrices. I am interested in the following ODE in $\mathbb{R}^n$ $$ \frac{dx_t}{dt}=Ax_t+Bx^+_t $$ where $x_t^+=(x^+_{1,t},...,x^+_{n,t})$ and $(\cdot)^+$ is the rectifier: $...
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2 answers
651 views

What is the current status on methods to find limit cycles?

What are the current best methods to show analytically the existence of a limit cycle in a $n$-dimensional system of the form: $$ \frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x}) $$ Where $...
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Stabilization of non-autonomuous 1-d wavs equation

I want to ask two questions about the stabilization of the equation $$\eqalign{ & {y_{tt}} = k(t,x){y_{xx}}+a(t,x){y_t}+ b(t,x){y_x}+ c(t,x){y_x} +d(t,x)y \ \ (t,x) \in {\text{ }}(0,\infty ) ...
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2 votes
0 answers
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Strong stability of the wave equation with time depending potential

It is well known that the wave equation with frictional damping $$\eqalign{ & {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr ...
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3 votes
0 answers
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Part 2: When does the maximal root of this polynomial have unit magnitude? Prove an inverse linear relation between parameters

Examine the polynomial $$ x^{\tau+1}-\left(1+m\right)x^{\tau}+mx^{\tau-1}+\left(1-m\right)\alpha=0\,. $$ with positive parameters $\tau,\alpha$ and $m<1$, and denote $\left|x_{\max}\left(\tau,\...
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4 votes
1 answer
116 views

When does the maximal root of this polynomial have unit magnitude? Prove an inverse linear relation between parameters

Examine the polynomial $$ x^{\tau+1}-x^{\tau}+\alpha=0\, $$ and denote by $\left|x_{\max}\left(\tau,\alpha\right)\right|$ as the maximal magnitude of a root of this equation. For $\tau>1$, I ...
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1 vote
0 answers
90 views

Bound for Expectation of Singular Value

In my case, $X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$ is a function of Rademacher variables $\boldsymbol{\delta}\in\{1,-1\}^M$ with $\delta_i$ independent uniform random variables taking ...
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3 votes
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conditions for asymptotic comparison to hold

I have the following simple dynamical system: \begin{align} x_1' &= a - f(x_2)x_1\\ x_2' &= bx_1 - cx_2, \end{align} where all parameters and initial conditions are positive. $f(x_2)$ is a ...
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0 answers
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Global stability question for system with a unique locally-asymptotically-stable steady state

I have an ordinary differential system of dimension 3 that contains a locally-asymptotically-stable unique fixed point. Additionally, the system is strictly-positively invariant and bounded. Now, ...
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7 votes
1 answer
353 views

How to study the global stability for this 3D system?

I am studying a biological system (HIV) and arrived at this simplified dynamical system: \begin{align} x_1' &= a_1 + a_2x_2 - a_1x_2 - a_4x_1 - a_5\frac{1+a_6x_3}{1+a_7x_3}x_1\\ x_2' &= a_5\...
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  • 493
0 votes
1 answer
57 views

Nonsmooth dynamical system (DAE) - systematic way to calculate period numerically?

What I have in mind is a mechanical system that is described by an implicit system of ODEs or a system of DAEs (differential algebraic equations). The system is asymptotically stable, meaning that ...
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0 votes
1 answer
102 views

Existence of a Lyapunov function for $-h'\varphi'+\varphi''$ where $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz

Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ The formal adjoint of $L$ is $$L^\ast\psi:=\psi''+(h'...
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1 vote
1 answer
210 views

Marginal stability of discrete linear time-invariant system

I have a question about marginal stability of a system: \begin{equation} \mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1] \end{equation} I would adapt the definition of marginal stability from this ...
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