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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Is it possible that a non-autonomous non-linear differential equation can be transferred into an autonomous? [closed]

Is it possible that a non-autonomous non-linear differential equation can be transferred into an autonomous? a) Give an example. b) what implications would it be to use one or the other scheme.
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Logistic model feasability implies stability [closed]

I have a model with logistic grown, as the lotka-volterra with limiting resources by carrying capacity. Does any one knows a theorem which says that in the logistic model, the feasability of the non ...
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Stability in the presence of vanishing inputs

Consider the system $\dot x= f(x)+u$, where $u=e^{-t}$ is a vanishing input. Let first $f=-x$: Then the Lyapunov candidate $V=1/2 x^2$ has the following time derivative: $\dot V= -x^2+xu\leq -x^2+|ux|...
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ODE: The Lyapunov stability of a set

When talking about the Lyapunov stability of a dynamical system, we usually take some point in the domain to test its stability. For example, for the ODE $\dot{x} = -x$, the origin $ x = 0 $ is ...
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How to conclude on the equilibrium point for the Jacobian matrix

I am trying to classify the equilibrium point of this system x' = $-2xy$ y' = $-3x^2 -y^2 + 4$ When I find the equilibrium points, I get $$(0,0) , (0,-2), (\frac{-2}{\sqrt3},0), (\frac{2}{\sqrt3},0)$$ ...
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Research of paper [closed]

I search for a paper where they study a stability of a model where the term source depends on $t$ only..In the papers that I have read before ,the othors neglect this term and I don't know if there ...
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Determine whether the solution of pendulum equation without friction is stable

Let us assume we have the simplest DE of a pendulum, without friction: $$\frac{d^2\phi}{d t^2}+\omega^2sin(\phi)=0$$ where $\phi$ is the angle of alteration. Boundary and initial conditions: $\phi(t=0)...
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Linear stability analysis of the nonlinear Schrödinger's equation

I have the following equation - $u_z = \frac{i}{2}u_{tt} + i|u|^2u$ where the subscripts denote partial derivatives with respect to the corresponding variable. I will be doing the stability analysis ...
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How to find a Lyapunov function in this case?

We have the system of differential equations $$ \begin{aligned} \frac{dx}{dt} &= y + \sin{x}\\ \frac{dy}{dt} &= -5x-2y. \end{aligned} $$ It's necessary to prove that the system is stable using ...
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Finding a Lyapunov function and proving the stability

I'm trying to find a Lyapunov function for $(0, 0)$ in the system \begin{cases} x' = 2x - 2y - (2x - y)^3\\ y' = 4x - 2y + (x - y)^3 \end{cases} I thought that the following one would ...
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Question about stability of non isolated equilibrium

Suppose I am studying the stability of a nonlinear system using the Lyapunov inverse theorem, and suppose I get a jacobian matrix that is singular. From the theory, I know that this implies that the ...
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Stability of stationary points of an ODE system

I have the next bidimensional system $$x'=x-2y+x^2-y^2$$ $$y'=-2x+y+3x^2-3y^2$$ The stationary points are $(0,0)$ and $(\frac{7}{8},\frac{5}{8})$, I tried construct a Lyapunov function but I'm stuck. ...
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How to analyze the derivative of a Lyapunov function

I am studying Lyapunov stability. My question is really short, and it is: Suppose I get a derivative of a Lyapunov function of the form: $\dot{V}(x,y)=-x^5-y^2$ what can I conclude about the stability ...
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PDE‘s stability

I am struggling with the following PDE's stability, my intuitive is to use the semigroup theory, but it seems that I can not even compute the eigenvalues, any other idea? $$w_t=-aw-b^2\int_o^{2\pi}...
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56 views

Stability of a degenerate equilibrium in a planer ODE using center manifold approach

I have a planer ODE system which is given by \begin{eqnarray} \begin{array}{lll}\begin{cases} \frac{dx}{dt} = p_{20}x^2+p_{11}xy+p_{30}x^3+p_{21}x^2y+p_{40}x^4+p_{31}x^3y+p_{22}x^2y^2,\\ \frac{dy}{dt} ...
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Finding the region of attraction using a Lyapunov function

I'm trying to find an estimate for the region of attraction of an equilibrium point. The notes from Nonlinear Control by Khalil suggest that defining $$ V(x) = x^TPx, $$ where $P$ is the solution of $$...
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What does $\operatorname{Tr}>0$ imply about complex eigenvalues of a Jacobi matrix?

I am currently taking a mathematical biology course and am working through the notes. We are covering stability analysis using the Jacobian matrix. One of the conditions for checking the stability is ...
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Are there Lyapunov stability conditions that focus on a subset of the available state equations?

Are there Lyapunov stability conditions that focus on a subset of the available state equations? Essentially, I have a system of nonlinear ODEs and I only care that some of the variables converge to ...
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Asymptotically stability of positive system

I currently study control theory and read about positive linear systems. In Lorenzo Farina and Sergio Rinaldi's book: "Positive Linear Systems: Theory and Applications", chapter 5 about ...
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Lyapunov equation and Time-varying homogeneous system

Consider the following time-varying homogeneous system $\dot{x}(t)=A(t)x(t)$. Assume that there exist an $n\times n$ symmetric matrix $Q(t)$ that satisfies $$\nu I\leq Q(t)\leq \rho I,$$ $$A^T(t)Q(t)+...
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Checking asimptotic stability of a critic point.

I've studied all the critical points of the following system using Hartman's Theorem, but for the (1,0) doesn`t apply because it's not hyperbolic. So I would like to find a strict Lyapunov function to ...
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Application on Gronwall's inequality

Using Gronwall' inequality, I need to show that the solution of the following initial value problem $x'(t)=(1-a \cos{t})x$, $x(0)=x_0$ satisfies $|x(t)| \leq |x_0| e^{(1+a)t} $. Here, $0<a<1$. ...
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About the truncation error of the scheme in the question

Question I have been trying to solve this messy question for a while. I think, to solve this, Taylor series of the unknown terms must be expanded around point (x_m, t_n) such as this: expansions Is it ...
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For a SISO LTI system, does being internally stable imply BIBO stability?

For a single-input single-output LTI system, $$ \dot x =Ax+Bu $$ $$ y =Cx $$ Does being internally stable, i.e. A is positive semidefinite, imply being BIBO stable?
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Stability of Explicit midpoint method

I am trying to determine the stability region of the well known explicit midpoint method $$y_{i+1} = y_i + h f\left( t_i + \frac h 2, \ y_i + \frac h 2 f(t_i, y_i)\right)$$ and after following the ...
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Proving Stability of a Function of Laguerre Polynomials

I'm trying to prove that the following potential is stable at its critical point: $$ F_{\textbf{n}}(x) = x - \sum_{\ell=1}^{r} \ln G_{n_{\ell}}(x), $$ where $\textbf{n} = (n_1, n_2, \ldots, n_r)$ and $...
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Stabilize a state of equilibrium

I'm having trouble solving a control theory problem. I have a differential equation: $$\dot{x}=-2x+Cx\gamma(t)$$ And I know that $|\gamma(t)|\le1$ Can I make the state of balance stable by choosing $C=...
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Stability of KdV equation

I want to find the stability of $$u_t + (1 + \pi^2)u_x + u_{xxx} = 0$$ Applying forward euler method with forward and central differenve schemes, I get $$ \frac{U_n^{j+1} - U_n^j}{k} + (1 + \pi^2)\...
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What is the classification of the bifurcation of a tent map?

Considering the tent map where $x_{n+1} = f(x_{n})$ and $f(x)$ is defined as $$ f(x)= \begin{cases} \mu x, & 0 \leq x\leq \frac{1}{2} \\ \mu - \mu x, & \frac{1}{2}\leq ...
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Convergence of monte-carlo methods.

Consider an algorithm to calculate $\pi$. Take a square $[0,1]\times [0,1]$ and make a quarter circle inside it of radius 1 centered at one of the corners. Now choose $N$ uniformly distributed random ...
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Stability of the equilibrium states

I have a function defined by the following differential equation $$ \frac{\mathrm{d}\varphi}{\mathrm{d}t} = \gamma - F(\varphi) $$ where $F(\varphi)$ is a $2\pi$-periodic function) and the chart of ...
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How to use Euler Theorem on homogeneous function to obtain $V' = -[2T - (\frac{\partial T}{\partial q}|q) - (\frac{\partial \pi}{\partial q}|q)]H$

Given the expression $V' = -[-(q|\frac{\partial H}{\partial q}) + (\frac{\partial H}{\partial p}|p)]H$ where (a|b) denotes the scalar product, and H is the hamiltonian ($H = T(q, p) + \pi(q)$, and ...
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Stability theory and Lyapunov functions

Does Global Asymptotic Stability imply Global Uniform Asymptotic Stability? What conditions need to be satisifed for both types of stability?
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existence of a specific solution to a difference equation

I want to find the necessary conditions for positive sequence of real numbers, $\{a_n\}$ for which the following difference equation has a solution where $0 < \theta_n < \frac{\pi}{4}, \forall n$...
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Given $V=(q|p)H,\,H(q,p)=T(q,p)+\pi(q)$, what is $\frac{dV}{dt}$?

I am studying the book "Stability Theory by Lyapunov's Direct Method", by Rouche, Habbets and Laloy, and in theorem 2.10 it defines the function $V=-(q|p)H$, where $(q|p)$ is the scalar ...
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Routh Table First Column 0: Total number of RHP poles

I was doing some research and found that when a first column 0 appears but everything else is not necessarily 0 in the row, then there exists poles with nonnegative real parts (or positive real parts ...
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If $B(q)$ is a symmetric matrix and $B(0)$ is positive definite, is $T = p^T B(q) p$ positive definite?

In book "Stability Theory by Lyapunov's Direct Method" by Laloy there is a General Hypotheses concerning stability, which says: "The kinetic energy is $T(q, p) = \frac{1}{2} p^TB(q)p$, ...
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Lyapunov stability f.o.differential equation

I have first order differential equation: $$y'-\frac{y}{x}+3y^2x=0$$ I found a general solution: $$y=\frac{x}{x^3+C_1}$$ $$C_1=\frac{x-x^3y}{y}$$ How can I check the stability of the solutions $y(1)=0$...
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Check if this system has any attractor

We have the following system: \begin{cases} h'=h(1-h-a_{12}p) & \\ p'=\rho p(1-p-a_{21}h) &, \rho>0, a_{12}>1, a_{21}\in (0,1).\\ \end{cases} We want to see if we have ...
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Unstable manifold of an unstable fixed point of a steepest descent iteration

Let $f:\mathbb R\to\mathbb R$ be differentiable $t:\mathbb R\to(0,\infty)$ be a "step size" function and $$\tau:\mathbb R\to\mathbb R\;,\;\;\;x\mapsto x-t(x)f'(x).$$ In the steepest descent ...
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Does there exist other version of Lyapunov-Like Lemma to solve this question?

Lyapunov-Like Lemma: If an energy function $\mathrm{V}(\mathrm{t}, \mathrm{x})$ satisfies the following conditions: $\mathrm{V}(\mathrm{t}, \mathrm{x})$ is lower bounded $\dot{\mathrm{V}}(\mathrm{t}, ...
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What is a suitable choice of Lyapunov function for this system?

I have the following nonlinear system: \begin{align}\dot x&=x(2-y^4) \\ \dot y &= -3x(1+y^3) \end{align} I first verified this system using linearization; however, the eigenvalues are $\lambda ...
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Asymptotic stability of compact sets

Let $\Omega$ be a topological space, $\tau:\Omega\to\Omega$, $A\subseteq\Omega$ and$^1$ $$E(A):=\left\{x\in\Omega\mid\forall N\in\mathcal N(A):\exists n_0\in\mathbb N_0:\forall n\ge n_0:\tau^n(x)\in N\...
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Equivalent definition of an “attractor”

Let $\Omega$ be a topological space, $\tau:\Omega\to\Omega$, $\operatorname{Orb}(\tau,U):=\bigcup_{n\in\mathbb N_0}\tau^n(U)$ for $U\subseteq\Omega$, $A\subseteq\Omega$ and$^1$ $$E(A):=\left\{x\in\...
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Relation between stability of fixed points of discrete and continuous dynamical systems

Let $f:\mathbb R^d\to\mathbb R^d$ be a diffeomorphism, $x_0$ be a fixed point of $$\frac{\rm d}{{\rm d}t}\varphi_t(x)=f\left(\varphi_t(x)\right)\tag1$$ (i.e. $f(x_0)=0$) and $A:={\rm D}f(x_0)$. It is ...
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Does a global attractor contain every (forward) invariant set?

In Remark 2.1.1 of this work, it is claimed that "the global attractor of a dynamical system contains every invariant set". It is not completely transparent what their definition of a "(...
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How can we show this set is stable, but not an attractor?

Let $\Omega$ be a topological space and $\tau:\Omega\to\Omega$. $A\subseteq\Omega$ is called stable if for every neighborhood $V$ of $A$, there is a neigheiborhood $U$ of $A$ with $$\tau^n(U)\subseteq ...
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Is every fixed point in an attractor stable?

Let $\Omega$ be a topological space and $\tau:\Omega\to\Omega$. We say that a fixed point $x_0\in\Omega$ of $\tau$ is stable if for every neighborhood $V$ of $x$, there is a neigheiborhood $U$ of $x$ ...
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Is a fixed point $x^\ast$ of a map $\tau$ with $|\tau'(x^\ast)|>1$ asymptotically unstable?

Let $\tau\in C^1(\mathbb R)$, $x_0\in\mathbb R$ and $$x_n:=\tau(x_{n-1})\;\;\;\text{for }n\in\mathbb N.$$ I was able to show that if $x^\ast$ is a fixed point of $\tau$ with $|\tau'(x^\ast)|<1$, ...
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Stable and unstable manifold of a system driven by the logistic map

Let $\lambda\in[0,4]$ and $$\tau(x):=\lambda x(1-x)\;\;\;\text{for }x\in[0,4].$$ Noting that $$\tau'(x)=\lambda(1-2x)=0\Leftrightarrow\lambda=0\vee x=\frac12\tag1$$ and $$\tau'(x)=-2\lambda\tag2,$$ it'...

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