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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0answers
33 views

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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61 views

how to show that the operator $\mathcal {L}$ has only one negative eigenvalue?

Consider the operator $\mathcal {L}: H^2_{per}([0,L])\subset L^2_{per}([0,L]) \longrightarrow L^2_{per}([0,L])$ given by $$\mathcal{L}(y)=w\cdot y''+(3\varphi-1)y, \; \forall \; H^2_{per}([0,L]),$$ ...
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53 views

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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52 views

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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22 views

stability of NLS solutions with $N$ nodes

I have a question regarding the stability of standing wave solutions to nonlinear Schrodinger equations (NLS) with $N$ nodes on the real line. In case $N=1$, the stability result is known due result ...
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19 views

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 ...
2
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1answer
171 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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39 views

(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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48 views

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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37 views

Quantitative estimate on continuity with respect to parameter of ODE

Let $V:\mathbb{R} \rightarrow \mathbb{R}$ be a bounded, $\text{Holder}$ continuousfunction with degree $\delta\in (0,1]$ such that $\inf_{\mathbb{R}} V = 0$ but $V(x) > 0$ for all $x\in \mathbb{R}$....
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59 views

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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1answer
87 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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1answer
170 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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1answer
84 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 knowlegde to solve it. Problem definition: Let $f(\xi) \in \...
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30 views

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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57 views

lyapunov stability for observer in dynamical system

I would like to use a stability analysis for the following problem: Having a system discrete time $$ \mathbf{x}(k+1) = \mathbf{Ax}(k) + \mathbf{B}\,\left(\mathbf{u}(k) + \mathbf{\widehat{z}^*}(k)\...
4
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1answer
91 views

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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2answers
237 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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0answers
32 views

property of a BIBO stable system

I'm working on a high dimension piece-wise linear system (PLS) which is already proved bounded-input, bounded-output (BIBO) stable, can I draw a conclusion that this system will always converge to ...
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17 views

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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19 views

Lyapunov stability is not reliant from start conditions on t

We have simple system of linear differential equations $\dot x = f(t, x)$, where $x = (x_1, x_2, ..., x_n)$, $f$ is defined, continious, satisfies Lipschitz conditions for $x$ in domain $G = (c, +\...
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47 views

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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59 views

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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1answer
106 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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0answers
72 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 ...
3
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0answers
61 views

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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54 views

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, ...
7
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1answer
269 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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1answer
39 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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1answer
99 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'...
1
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1answer
136 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 ...
3
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1answer
144 views

Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?

Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...
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77 views

Stability of dynamical system via Lyapunov

I have a dynamical system which has the following form: $\dot x=\mathcal F_1(m_1)x+\mathcal F_2(m_2)x$. My objective is to find the parameters $m_1$ and $m_2$ via LMI (linear matrix inequality) using ...
3
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1answer
251 views

Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question. In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator? Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space. ...
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1answer
257 views

Von Neumann stability vs eigenvalues of amplification matrix

My, limited, understanding of the stability analysis of PDEs is that broadly speaking there are two methods: von Neumann analysis which looks at the growth the error of the solution, as described here ...
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1answer
225 views

Center manifold theorem and case of all zero eigenvalues

Is the center manifold theorem applicable if say for a planar (2D) system of non-linear ODE, the stability matrix has both eigenvalues zero? Of course, there is only one eigenvector. If not, what is ...
7
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1answer
686 views

(In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations $$ \dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\ \dot{x}_2(t) = -\gamma x_2(t) - \cos(\...
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1answer
125 views

Boundary controllability of the heat equation and observation

I'm studying the boundary controllability of the heat equation \begin{array}{c} y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\ y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\ u(...
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1answer
130 views

Stability of eigenvectors for diagonal perturbations

In a previous question I asked about the stability of eigenvalues with respect to diagonal perturbations. Following results from the book Matrix Analysis (by Roger A. Horn & Charles R. Johnson) ...
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0answers
66 views

Showing a modified system of quadratic equations is stable

I have and $n$ dimensional dynamical system, given by $\dot{x} = M D(x) P x - \frac{c}{2}x$ $P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...
2
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1answer
118 views

On local attractivity of a coupled non-linear differential equation

Consider a dynamical system described by the following coupled non-linear differential equation \begin{align} \dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\ \dot{x}_2(...
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0answers
157 views

Stability when linearization fails

The dynamics of the $j$th system: \begin{equation} \begin{split} \dot{\overline r}_j &= h (\overline r_j) \,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...
4
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1answer
257 views

Almost sure stability of a scalar, nonautonomous, nonlinear SDE

I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it. Maybe there is someone here who can either close the gap in one of the existing answers or has ...
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0answers
150 views

Projective unitary flat structures of $\mathbb{P}^1$-bundles on Riemann surfaces

Narasimhan and Seshadri proved a rather surprising result about vector bundles on a compact connected complex manifold $X$. That is Two holomorphic vector bundles arising from unitary representations ...
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0answers
58 views

Exponential Convergence Under a Lyapunov-Like Assumption

Consider $V=\mathbb{R}^d\times\mathbb{R}^n$ with coordinate $x^T=[\theta^T,\sigma^T]$. I have an ODE of the form: $\dot{x}=F(x)$, where $F$ is assumed to be sufficiently smooth. Suppose that there's ...
52
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4answers
4k views

Is there a mathematical and information theoretic explanation for this cube packing phenomenon?

I saw this unintuitive result on dice packing: A jumble of thousands of cubic dice, agitated by an oscillating rotation, can rapidly become completely ordered, a result that is hard to produce ...
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0answers
79 views

Stochastic stability of “open” continuous-time stochastic systems: reference request

I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but ...
2
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0answers
78 views

Asymptotic stability of eigenvalues by compact perturbations

I need some references concerning the asymptotic stability of eigenvalues by compact perturbations. In [T. Kato, Perturbation theory for linear operators] there are some results concerning stability ...
3
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2answers
372 views

Relation between controllability and stability of PDE

In general, when we talk about controllability, we talk about proving the existence of a control input that transfers the state to a desired state at a desired time $T$. However, when we talk about ...
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0answers
60 views

Ratio dependent predator prey model

In the article on Global qualitative analysis of a ratio-dependent predator–prey system- Kuang, 1998 The system is where a, K, c, m, f, d are positive constants that stand for prey intrinsic ...