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Questions tagged [stability-theory]

Stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.

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Asymptotic stability and Lyapunov functions

I fail to understand a passage in the proof of the following theorem (right after the definition that gives the context of my question): (Definition of Lyapunov function) Let $\Omega$ be a ...
ebenezer's user avatar
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Given is the system of differential equations

Given is the system of differential equations: $$\begin{cases}\dot x=4y \\ \dot y=-3x \end{cases}$$ (a) Write the first integral of the system. Is the system conservative? Explain. (b) Sketch the ...
GENERAL123's user avatar
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Mathematic Modeling [closed]

How to solve the following equation manually? I want to find the equilibrium point of the system of equations below https://drive.google.com/file/d/1gPUU3N2m1b4tuhxUHaRFHRaOv8gll7_c/view?usp=sharing
thesatria's user avatar
2 votes
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How to simplify the following stability criteria?

I'm trying to understand the mathematics explained in the following video. Basically, we want to identify the constraints on the parameters $k_1$ $k_2$ and $T$ to have a stable second-order system ...
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Lyapunov stability of a periodic system

Consider a planar system $$ \begin{cases} \dfrac{\mathrm{d}x}{\mathrm{d}t}=-y,\\ \dfrac{\mathrm{d}y}{\mathrm{d}t}=(a+\varepsilon\cos t)x, \end{cases} $$ where $a>0$ and $a\notin\{\dfrac{n^2}{4}\ |\ ...
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Asymptotical stability and stability for homogenous ODE

Let $A \in \mathbb{R}^n$. Prove, that the zero solution of $x'=Ax$ is Asymptotically stable $\iff$ $Re(\lambda)<0$ for all eigenvalues $\lambda$ of matrix A Stable $\iff$ $Re(\lambda) \leq 0$ and ...
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Stability of Hamiltonian system on degenerate critical point.

I'm trying to find information on the stability of the following ODE: $$ x'' = x^4-x^2.$$ We know that it has a Hamiltonian $H(x,y) = \dfrac{y^2}{2} - (\dfrac{x^5}{5} - \dfrac{x^3}{3})$. The orbits ...
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Connection between Liouville's formula and stability of linear systems

In my university textbook I have a statement: For a linear differential system $\frac{dx}{dt}=A(t)*x$ to be stable is necessary that $\int_{0}^{t} SpA(\tau)d\tau <= M$, where $M$ is a constant, $...
Snork Maiden's user avatar
1 vote
1 answer
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Stability of stationary points

In the article by Hirsch "On stability of stationary points of transformation groups It's mentioned that $0$ is a stable stationary point of the diffeomorphism $f(x)=x+x^3$ (stationary point of ...
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Question about the proof that uniform asymptotic stability can be characterized by KL function. (Lemma 4.5 in Nonlinear Systems (3rd) by Khalil)

Lemma 4.5 in Nonlinear Systems (3rd): Consider the nonautonomous system \begin{equation} \dot{x} = f(t,x) , \end{equation} where $f : [0,\infty) \times D \to \mathbb{R}^n$ is piecewise ...
Lau's user avatar
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Exercise about stability and inequality in ODE.

I am stuck with this exercise. Let \begin{equation} x''(t) + x(t) = \epsilon \sin(x(t)) \end{equation} with initial conditions $ x(0) = x_0, \ x'(0) = v_0 $. 1. Write the problem as a first ...
M159's user avatar
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Confusion about stability of PDEs.

I have been reading about Stability Theory and have been left with some questions at is seems to me that some of its notions are not very well-defined or at least inconsistently used. Consider the ...
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Lyapunov stability in a one-sided neighborhood?

Consider a switching system $$ \dot x = { - x, \quad {\rm{if}}\quad x \ge 0} $$ $$ \dot x = {v \left( t \right), {\rm{ if}}\quad x < 0} $$ where $ v(t) $ is bounded but indefinite (can be ...
Tag's user avatar
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Studying stability of pde

I have a problem with studying the stability of this PDE. $$U_t = U_{xx} + f(U).$$ Let $U^{*}(x)$ be a solution for this equation. As conditions we have ${U^{*}}'(x) > 0$ for $x < x_{0}$, ${U^{*}...
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Stability estimate for elliptic reaction-diffusion problem with inhomogeneous Neumann boundary conditions?

I want a stability estimate for the problem $$-\Delta u + u = f \text{ on } \Omega,$$ $$\nabla u \cdot n = g \text{ on } \partial \Omega,$$ where $g \in L^2(\partial \Omega)$. The problem has ...
1Teaches2Learn's user avatar
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Please help with literature for Lyapunov stability for non-linear observer

I'm working with an approximation-based observer design for reaction-diffusion PDE, where I apply the Petrov-Galerkin approximation to a non-linear PDE and get the following ODE: $$\dot{\beta} = A\...
Áron Fehér's user avatar
1 vote
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particle in motion under the influence of friction

Let's consider a particle of mass = 1 Kg moving according to the law $$ \ddot x(t) = -V'(x(t))-\frac{2}{3}\dot x(t) = -x(t)^3+x(t)-\frac{2}{3} \dot x(t). $$ (The potential energy is $V(x)=\frac{x^4}{4}...
dattiluca's user avatar
2 votes
1 answer
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Prove stability of solution to a system of differential equations

How to prove that the zero solution of the following system of differential equations is stable, but not asymptotically stable? \begin{cases}\dot{x} = y - y^2 \\\dot{y} = -x\end{cases} It is easy to ...
ElectroSchOOp's user avatar
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Reproducibility of the Asymptotics of ODEs

My question has its origin and is related to this problem in heat conduction but is more general in scope. Let $\mathbf {f(x, u)}$ be a sufficiently smooth function of variables $\mathbf x \in \...
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2 votes
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Simple way in detecting astable solution from Differential Algebraic Equation

I have a semi-explicit DAE (Differential Algebraic Equation) with the following form: $$ \begin{align} \mathbf{\dot{s}} &= \mathbf{f}(\mathbf{s}, \mathbf{x}) \\ \mathbf{0} &= \mathbf{g}(\...
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Derive equation unstable manifold

I am studying the book Differential Equations and Dynamical Systems (third edition) of Perko and in problem set 2.7, Question 4/5 they give the following system: \begin{align} \dot{x}_1&=-x_1\\ \...
Cathematics's user avatar
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Detailed exp does Euler's method fail for second order sinusoidal ODE?

I am trying to explain, for a seminar, why this ODE fails with Euler's method. $$ \partial^2_x y= \sin(x) $$ for boundaries, $L=2\pi$, $y(0)=y(L) = 0,\ y'(0) = y'(L) =-1$, which has exact solution $$ ...
K-Q's user avatar
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Approximating solution of a second degree ODE

Consider the second-order DE $y'' + p(x) y = 0$, such that $\int_{}^{\infty} t|p(t)| dt < \infty$. Show that, for any solution $y(x)$, $\lim_{x\to\infty} y'(x)$ exists, and every nontrivial ...
R_Squared's user avatar
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how to show global asymptotic stability with $V(x)=f(x)^{T}Pf(x)$ as a lyapunov function.

consider the system $f(x)=\dot{x}$ with $f(0)=0$, $f(x)$ is continuously differentiable. $f(x)$ can be written as $f(x)=\int_{0}^{1}\frac{\partial f}{\partial x}(x\sigma)x\partial\sigma$ (The first ...
TiredMechanicalEng's user avatar
1 vote
1 answer
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Doing the characteristic polynomial I can't move forward, where do I fail?

I have this problem: Consider the following system of equations $$ \begin{align*} y_{1}'(t) &= y_{3}, \\ y_{2}'(t) &= -3y_{1}, \\ y_{3}'(t) &= \alpha y_{1} + 2y_{2} - y_{3}. \end{align*} $$...
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How to ensure stability of 2nd order linear non homogeneous ODE?

Consider the following second order linear ODE, $$L[g]=f(x)g''+b(x)g'+c(x)g=a,$$ where $a\neq 0$ is a constant and the coefficients can be as regular as you like. I am interested in conditions on the ...
Emmet's user avatar
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computational complexity of stabilization problem in Boolean control network

Stabilization problem is a fundamental problem in control theory. There are many literature to achieve stabilization but fewer results are related in computational complexity. Consequently, we ...
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Why is it only possible to build a homeomorphism rather than a diffeomorphism in Hartman-Grobman Theorem"

It is emphasized in all the materials that I have seen about Hartman-Grobman theorem (including textbooks and Wikipedia) that the topological conjugacy is only a homeomorphism but not diffeomorphism, ...
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Example of non-hyperbolic fixed points and their stability

Recently I have started studying (1-dimensional) dynamical systems and the first thing that I came across fixed points, specifically hyperbolic and non-hyperbolic fixed points. What would be an ...
variableXYZ's user avatar
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6 votes
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Can an ODE system never converge to its stable equilibrium in the long run?

I have the following coupled non-linear ODE system, which describes a biological system: $$ \begin{cases} \dfrac{dp}{dt} = -\gamma p f,\\ \\ \dfrac{df}{dt} = \gamma p f,\\ \\ \dfrac{dT}{dt} = \left( 1 ...
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1 answer
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Limit of an autonomous ODE

Given the differential equation $$\frac{dy}{dt}=y(y-1)(y-3)$$ I found $y=1$ is a stable point while $y=0$ and $y=3$ are unstable. However, I have to evaluale $$\lim_{t\to\infty}y(t)$$ without solving ...
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ROC of a fractional order system in respective to its poles

My understanding about fractional order system is that it can have poles on the right hand side of imaginary axis in s-plane and yet being stable. (statement 1) There are other theorems about ...
Anders's user avatar
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2 votes
1 answer
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Stability of the null solution of a differential equation

Is there a differential equation: $$\ddot x(t)=f(t, x(t)),\ f\in C^1(\mathbb{R}^2), \forall t\in \mathbb{R}\ f(t,0)=0;$$ that for any solution x(t) the following is true: $$\forall \epsilon>0\ \...
Stanislav Lavrov's user avatar
4 votes
1 answer
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Is a system, that is globally asymptotically stable for any constant input also input-to-state stable? [closed]

I am referring to the ISS definition by Sontag of ${\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}$ I understand that 0-GAS is a necessary condition for ISS. But is GAS for all ...
LCG's user avatar
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Reference Request: Stability of solutions of forced PDEs

Given a system of ODE or PDE and a solution $u$ such that $$ \partial_t u = \mathcal{L}u , \quad u(t=0 ) = 0.$$ for some operator $\mathcal{L}$, one can ask the following question Let $u$ be the ...
newbie's user avatar
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How could I determine the stability of these equilibrium points if $\lambda > 0$?

We consider the uniparametric differential equation $$\frac{dx}{dt} = \left(\lambda - x^2 + 2x\right)\left(\lambda - x^2\right).$$ I want to determine the equilibrium points as well as their stability ...
Cyclotomic Manolo's user avatar
3 votes
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Divergence of a dynamic system after element sign change

Suppose there is a continuous dynamic system of order $n+1$ given by $$\begin{align} \dot{x}_1 &= Ax_1 + F(t)x_1 + G_1(t)x_2 \\ \dot{x}_2 &= kx_2 + G_2(t)x_1 \end{align}$$ where $x_1\in\...
Raddeo's user avatar
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Mapping the zeros of a transfer function in S domain and Z domain

The $s$ and $z$ domain variables are connected through the expression $z = e^{sT}$, in which, $s$ is the Laplace variable and $T$ is the sampling period. However, I have found no rigorous method that ...
Saeed's user avatar
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2 votes
1 answer
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Lyapunov Asymptotic Stability

I am still new to Lyapunov stability and I have a question: The system is: $\dot{x}_1 = x_2(1-x_1^2)$ and $\dot{x}_2=-(x_1+x_2)(1-x_1^2)$ I used $V(x)=\frac 1 2(x_1^2+x_2^2)$ Then, I get $\dot{V}(x)= -...
Johny's user avatar
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Real part of the eigenvalues of a matrix product

Given a real square matrix $A \in \Re^{n \times n}$, let's say its eigenvalues have a lovely property: $Re(\lambda^A_i) < 0$. We also have another diagonal real square matrix $K \in \Re^{n \times n}...
lostintimespace's user avatar
3 votes
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Ruling out finite time explosions to infinity of Riccati differential equations

Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions. Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive ...
cfp's user avatar
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Identifying Bifurcation

I am trying to identify bifurcation of my $3D$ system near $a=6.58$. I am getting trajectories as shown in the picture and I am guessing it is Saddle-Node periodic orbit bifurcation since I am getting ...
SHR's user avatar
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1 vote
1 answer
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Linearization of a nonlinear third order ODE and stability [closed]

I would like to know if the following differential equation ($\alpha,\beta,\gamma,d,\Lambda,w$ are constants) $x'''(t)=\frac{1}{24 (3 \alpha -\beta )}\frac{x(t)^{-3 w-2}}{x'(t)} \left(36 \alpha x(t)^{...
Axionlike particles's user avatar
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Stability of discrete-time dynamical systems using Lyapunov stability where A is function of optimization variable

Hi I am trying to solve a constrained optimization problem using the Lyapunov stability. In the problem we aim to find $\beta$ such that $$\min_\beta ||\beta^TF-y|| \quad \text{s.t.}\quad A^{T}PA-P&...
geo200's user avatar
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1 vote
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Chetaev theorem for discrete time

In reading the following article: https://www.researchgate.net/publication/262736434_The_Chetaev_Theorem_for_Ordinary_Difference_Equations Theorem 1 seems to prove a discrete-time analog of Chetaev ...
xyz's user avatar
  • 1,000
2 votes
1 answer
87 views

Convergence with increasing Lyapunov function

Given a (autonomous) dynamical system, one can prove instability of a point via the Lyapunov method, by simply finding a Lyapunov function that increases in a neighbourhood of the point. This ensures ...
xyz's user avatar
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0 votes
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Singularity of a non- linear second order ODE

I have the encountered a singularity in the equation below . $$ y^{\prime \prime}(x)+\frac{2}{x} y^{\prime}+\left[y-\left(1+\frac{2}{x^2}\right)\right] y(x)=0, \quad 0<x<+\infty, $$ with ...
SR9054505's user avatar
2 votes
0 answers
59 views

Radially bounded Lyapunov function and global stability

I came accross this link about the necessity of the Lyapunov function being radially unbounded. My understanding is that this condition is unnecessary if the time derivative along solution ...
Yonatan's user avatar
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1 vote
1 answer
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Lyapunov Stability class K functions

I'm reading the book on Nonlinear System Analysis by M. Vidyasagar. I see they define functions of class K as continuous strictly increasing functions such that $\phi(0)=0$ and from there, they define ...
user1880062's user avatar
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Question on Lasalle's Invariance Principle

Consider system $\dot x = f(x)$ and let $\Omega\subset D$ be a positive invariant set. Let $V: D \rightarrow \mathbb R$ be a radially unbounded, positve definite function. Let the derivative fulfill $$...
Trb2's user avatar
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