Questions tagged [stability]
Stability theory, including global stability (in dynamical systems, where it can notably be used in combination with ds.dynamical-systems)
162 questions
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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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Stability Problem in a Nonlinear Dynamical System
Consider the nonlinear dynamical system given by the following differential equations
\begin{cases}
\dot{x} = y, \\
\dot{y} = x - x^3 - \gamma y + \delta x^2 y.
\end{cases}
I want to demonstrate that, ...
- 31
5
votes
1
answer
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Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 141
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votes
1
answer
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Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
- 1,714
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vote
0
answers
52
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Stability of Euler discretization
I am looking at the discretization of an ODE:
$$x_{n+1} = x_n + \alpha f(x_n),$$
where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...
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3
votes
0
answers
107
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Stability of a nonlinear dynamical system with non-elementary dynamics
I am trying to prove stability and get a non-asymptotic upper bound on the convergence rate of a nonlinear discrete-time dynamical system, whose dynamics are stated in terms of the (non-elementary) ...
- 31
2
votes
1
answer
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$R^1\Gamma = 0$, and the Mumford stability
Let $S$ be a smooth projective surface with an ample divisor $H$ so that $K_S \cdot H < 0$.
Let us consider the Mumford slope $\mu(E) = \frac{H \cdot c_1(E)}{\operatorname{rk}(E)}$ on $\...
- 5,857
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Characterization of multifunctions with globally asymptotically stable minimal invariant sets
Let $(X, d)$ be a compact connected metric space. Consider a compact-valued upper semicontinuous multifunction $F: X \rightrightarrows X$. The reachable set $R[x]$ of $F$ from $x\in X$ is defined as:
...
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The Frobenius integrability of distrbution and Hyers–Ulam–Rassias stability
Let $M$ be a compact Riemannian manifold. The norm of vector fields are computed with respect to the metric. Moreover for every distribution $D$, the orthogonal projection on $D$ is ...
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votes
1
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Norm bound in simultaneous stability to semidefinite program
In the context of robust control, I remember hearing that the two following problems are equivalent.
Find $P \succ 0$, such that $A P + P A^{\top} \prec 0$ for all $A \in \mathscr{A}$ where $$\...
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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)...
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0
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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 ...
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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 ...
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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 ...
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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 ...
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votes
2
answers
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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(\...
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3
votes
2
answers
339
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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 ...
- 10.3k
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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 ...
- 6,367
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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 = \...
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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 ...
2
votes
1
answer
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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}...
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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 ...
- 1,284
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votes
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answers
67
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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 ...
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votes
2
answers
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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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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 ...
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1
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1
answer
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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 ...
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2
votes
2
answers
349
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dp-minimality and stability
What are some of the common popular stable theories that are known to be dp-minimal (or not dp-minimal)?
Some dp-minimal examples I am aware of are strongly minimal theories, superstable theories of ...
- 151
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answers
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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 ...
- 373
3
votes
1
answer
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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 ...
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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
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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 ...
- 1,167
3
votes
1
answer
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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)&...
- 128k
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0
answers
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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_{...
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votes
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answers
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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 ...
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108
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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)}...
- 1,759
5
votes
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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 ...
- 6,045
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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)), \;\; t \geq 0, \; \;\boldsymbol{f}(\boldsymbol{0}_n) = \boldsymbol{0}...
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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 ...
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vote
1
answer
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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 ...
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vote
0
answers
25
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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 ...
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votes
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453
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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\...
- 6,367
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(In)stability of a two-dimensional dynamical system
Consider the following system of coupled differential equations
\begin{eqnarray*}
\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)...
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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
1
answer
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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 ...
- 8,336
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votes
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answer
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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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votes
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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, ...
- 6,571
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1
answer
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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 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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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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