# Questions tagged [stability]

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

110
questions

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votes

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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 "...

**-1**

votes

**0**answers

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
$$\mathcal{L}(y)=w\cdot y''+(3\varphi-1)y, \; \forall \; H^2_{per}([0,L]),$$
...

**1**

vote

**0**answers

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{...

**5**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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}
\...

**2**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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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votes

**0**answers

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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vote

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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 ...

**2**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**1**

vote

**1**answer

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 \...

**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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: $...

**4**

votes

**2**answers

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 $...

**0**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 ) ...

**0**

votes

**0**answers

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, +\...

**2**

votes

**0**answers

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
...

**3**

votes

**0**answers

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,\...

**4**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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\...

**0**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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**

vote

**1**answer

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**

votes

**1**answer

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(...

**2**

votes

**0**answers

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**

votes

**1**answer

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.
...

**0**

votes

**1**answer

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
...

**1**

vote

**1**answer

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**

votes

**1**answer

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(\...

**1**

vote

**1**answer

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(...

**0**

votes

**1**answer

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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vote

**0**answers

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**

votes

**1**answer

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(...

**1**

vote

**0**answers

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**

votes

**1**answer

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 ...

**5**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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**

votes

**4**answers

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 ...

**2**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**2**answers

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 ...

**1**

vote

**0**answers

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 ...