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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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Lyaponov function in dynamic system (in polar cordonates)

Let the following system in $R^2$ \begin{equation} (S) \left\{ \begin{array}{l c } \overset{.}{\rho}=\rho(1-\rho) \\ \overset{.}{\theta}=\sin^2(\frac{\theta}{2}) \end{array} \right. \end{equation} ...
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Von Neumann stability analysis - almost finished exercise

I wish to analyze the stability of the FTCS scheme for the equation $u_t = iau_{xx} + cu$ where $a \in \mathbb R, c \in \mathbb C$. I will succinctly go through what I did and where I'm stuck. A ...
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1answer
65 views

Requirement of Lyapunov Stability in Asymptotic Stability

In my Differential Equations course, we defined the equilibrium point $x_0$ of a dynamical system $\frac{dx}{dt} = f(x(t))$ (for $f$ defined on an open subset of $\mathbb R^n$, say $\mathbb R^n$ ...
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Compare solutions in some moment of time for two equations by Lyapunov derivatives?

We have two differential equations $\dot x_1=f_1(x_1)$ and $\dot x_2=f_2(x_2)$ which are too complex to solve but we could show by the same Lyapunov function $V$ (and its derivatives) that equilibrium ...
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What are hydrodynamic and non-hydrodynamic modes in the Navier-Stokes equations and how do they relate to stability?

I'm new here. I have been reading articles on Navier-Stokes equation and the stability analysis by studying different modes. However I am still confused by what is meant by modes as in the context of ...
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Multistep method

I need to work out the set values of the element $\alpha$ for which the three-step method below is stable. $y_{n+3}-(\alpha+1)^2y_{n+2}+\alpha(\alpha^{2}+2\alpha+2)y_{n+1}-\alpha^{2}(\alpha+1)y_{n}=hf(...
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Does Lyapunov Stability imply Attractivity for intervals on the real line?

For intervals on a real line, I have found a result which states that for a continuous map, attracting fixed points are Lyapunov stable. However, I found no result about the converse. So, is the ...
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How to Generate a Bifurcation Plot for a Recurssive Difference Equation

I am trying to generate a bifurcation plot for the following system (a single oscillator given by the difference equation) $x(n + 1) = f(x(n))$, with $f(x) = 1 - ax^2$, where $a$ is a constant ...
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1answer
23 views

Types of equilibria of a chaotic system

Is it correct to use the Jacobian matrix in determining the types of equlibria of a non-linear chaotic system of smooth ODEs? If not, is there a general approach?
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How to understand assumption from stability theorem?

Consider $$x'(t)=A\cdot x(t)+f(t,x(t)),\,\,x(0)=a$$ for $t\geq0$. Assume that $A$ is real $N\times N$ matrix with negative real parts of eigenvalues, $f\colon \mathbb{R}\times \mathbb{R}^N\to\mathbb{R}...
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Critical points of the following ODE

Consider the following nonautonomous, nonlinear ODE: $$y'(t)=\rho(W(t)y(t)+b(t)), \hspace{6pt}t\geq 0,$$ where $y(t),b(t)\in\mathbb{R}^{n},W(t)\in\mathbb{R}^{n,n}$ and $\rho:\mathbb{R}\to\mathbb{R}$ ...
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Stability of the following ODE

Consider the following nonautonomous, nonlinear ODE: $$y'(t)=\rho(W(t)y(t)+b(t)),$$ where $y(t),b(t)\in\mathbb{R}^{n},W(t)\in\mathbb{R}^{n,n}$ and $\rho:\mathbb{R}\to\mathbb{R}$ is some nonlinear ...
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59 views

How fast does a solution approach an equilibrium?

I have an autonomous dynamical system $\dot{\mathbf x}(t)=\mathbf f(\mathbf x(t))$ on $\mathbb R^2$, and I found that solutions are future asymptotic to an equilibrium point $\mathbf a$, i.e. $\lim_{...
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28 views

Globally uniformly asymptotic stability proof

In dynamics theory, here is a question A large class of time-varying capacitor-linear resistor networks can be described by equations of the form $$x'_i=-\sum_{j=1}^n[a_{ij}\,d_{1j}(t)+b_{ij}\,d_{...
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1answer
36 views

Stability of non-hyperbolic critical point in two dimensions, where the linearization has one zero egenvalue

I am having a two dimensional autonomous system $S' = 2S^3+2S^2+\frac{1}{2}SA-\frac{3}{2}S-\frac{3}{4}A$ $A' = 4AS^2+A^2+4AS$ which exhibits a critical point at the origin $(S,A)=(0,0)$, and others....
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Stability of equilibrium points of system of differential equations

Given $$\ddot{x}+\lambda\dot{x}=x-x^3,$$ which I rewrote to \begin{align} \dot{x}&=y \\ \dot{y} &= -\lambda y + x - x^3, \end{align} For the exercise we take $\lambda\geq 0$. I have determined ...
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Creating equilibria by adding a time-periodic term to an ODE?

Let $f\colon \Bbb R^N\to \Bbb R^N$, $g\colon[0,\infty)\to\Bbb R^N$ and consider the ODE $$ \dot x(t) = f(x(t)) + g(t), $$ where space- and time-dependence are additively separated. Is it possible to ...
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Complex dynamical system: check stability of point numerically

Given a dynamical system described by the equations (for i=1,...,N) $$\frac{d y_i}{dt} = P_i - by_i + K \underset{i \neq j}{\sum_{i=1}^N} \sin(x_i-x_j)$$ $$\frac{d x_i}{dt} = y_i$$ Say that I have ...
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Direction Field in MatLab

I need to plot the direction field and phase plane of the following ODE using matlab. I've tried using meshgrid w/ the quiver function, however, I'm not getting the correct field. I need to plot $\...
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1answer
29 views

Finite difference stability problem

My apologies for the title I'm not quite sure how to title a problem like this. I need to show the following result: $$u_j^{n+1} = e^{\Delta t\partial/\partial t}u_j^n$$ Where $u_j^n$ is the ...
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1answer
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Estimate in proving that linear instability implies nonlinear instability

I am trying to prove the exercise on page 3 of http://depts.washington.edu/bdecon/workshop2012/g_stability.pdf. This question was already asked before here: Linear instability implies nonlinear ...
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Eigenvalues of a block matrix and stability of a linear system of differential equations with time varying coefficients

Let $X \in \mathbb{R}^{n \times 1}$ and $P_1,P_0(t) \in \mathbb{R}^{n \times n}$ symmetric and positive definite for all $t \in \mathbb{R}$. Consider the dynamical system: $$ \ddot{X} + P_1\cdot \dot{...
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Checking the stability of finite difference schemes

How would I go about checking the stability of the following schemes. Usually I use Von Neumann Stability analysis however I'm not sure how this works as I have x and y and t dependence and I've only ...
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28 views

Prove a Lemma Involving Asympotically Stability

I am trying to prove the following Lemma: Lemma: Suppose that the point $x^*$ is a fixed point of $x(n + 1) = f(x(n))$ (1) while also an asymptotically stable(unstable) fixed point with respect ...
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31 views

Solving system of differential equations with unknown eigenvalues

I have 4 differential equations and a characteristic polynomial like $ λ ^4+ \frac{w^2*λ^2}{ɛ} - \frac{2kw^2}{ɛm}=0 $ where I denoted $ɛ$ as small deviation approximately zero, $m$: mass, $w$: angular ...
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1answer
41 views

Can we perturb a low rank map to a full rank map in a smooth way?

Let $f:\mathbb{R}^n \to \mathbb{R}^n$ be smooth. Can we find, for every $\epsilon>0$, a $C^1$ map $\tilde f:\mathbb{R}^n \to \mathbb{R}^n$ of full rank such that $\|df-d\tilde f\|_{C^0}<\...
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Different kinds of stability that apply to planar periodic orbits, and what do they mean?

This is a question about terminology related to orbit stability. I had wanted to ask about stability of orbits described in the paper Three Classes of Newtonian Three-Body Planar Periodic Orbits ...
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Counterparts to Schwarz matrices

Schwarz matrix (see Definition $3.1$) is a class of tridiagonal matrices which has special properties: the inertia $\{n_0, n_+, n_{-}\}$ of a Schwarz matrix is completely determined by the sign of the ...
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Is the Lorenz system well-posed in the Hadamard sense?

Apologies if this has already been discussed, but I searched the site and I couldn't find an answer. For the sake of simplicity, consider only ODEs, possibly depending on some vector of parameters $\...
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About topological conjugacy

The Smale horseshoe map $f$ is desribed in this page: What's the point of a Horseshoe map? A striking feature of this system is the stability of its dynamics: given any diffeomorphism $g$ ...
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1answer
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Finding the absolute stability

I am trying to find whether the following is stable absolutely using the improved Euler and the Adams-Bashforth 2 scheme, $u'=\begin{bmatrix} -20&0&0\\ 20&-1&0\\0&1&0\end{...
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Stability of an elliptic PDE

I'm reading An introduction to semilinear elliptic equations of Thierry Cazenave. In the middle of the text, he asserts that in general the groundstate solution (that is, the minimal solution with ...
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Confusion about Numerical Stability

Numerical Analysis is giving me some trouble. In specific, I'm highly confused by our definition of numerical stability. I'm hoping some of you can help me clear my confusion. Definition. An ...
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1answer
49 views

Does $x_{t} = 1-x_{t-1}$ have a stable steady state solution?

At steady state, $x = x_{t} = x_{t-1}$. So I can solve for the steady state value of $x=0.5$. The general rule of determining the stability of the steady state is that the $|\text{slope}|<1$. But ...
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Behavior of the iterated map $x_n=C-1/x_{n-1}$ when $|C|\leq 2$ [closed]

I am interested in analyzing the behavior of the map $$x_n=C-\frac{1}{x_{n-1}}$$. where $|C|\leq 2$. Here the $x_n$ are allowed to be complex. Obviously there are two fixed points at $x=\frac{C\pm\...
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1answer
42 views

Stability region for two step Nyström method

I have the following two step Nyström method: $u_{k+1}=u_{k-1}+2h\cdot f(u_k)$. I want to know the stability region, so I wrote this as $w_{k+1} = A\cdot w_k$ with $A=\left(\begin{matrix} 2h\lambda &...
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2answers
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Stablility of a linearized time-delay system

I have a linearized time-delay system as follows: $$\frac{\mathrm d X}{\mathrm d t} = a[X(t)-X^*] + b [X(t-R) - X^*], $$ where $a$, $b$ are constants, $R$ is the constant delay, and $X^*$ is the ...
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1answer
22 views

Stability of a linear equation

If $A$ is a matrix, then $e^{At} \leq C e^{-\lambda t}$ if and only if the spectrum of $A$ consists of eigenvalues with negative real parts. Is there a similar result, relating stability to the ...
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1answer
39 views

Stability of non-homogeneous ODE

I try to examine stability of non-homogeneous ODE system: \begin{cases} Dy_{1} = y_{1}+2y_{2} +\frac{3}{x^4} \\ Dy_{2}= 3y_{1}+4y_{2}+ \frac{3}{x^4} \end{cases} I tried to find solutions of such ...
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Is it possible for hamiltonian systems to have asymptotically stable rest points?

Is it possible for hamiltonian (dynamical) systems to have asymptotically stable rest points? By Liouville's theorem I got that for symplectic hamiltonian systems only Lyapunov stability is allowed. ...
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Obtain phase portrait of this system

I have a system which consists on a deposit with water, where the relation between the in and out fluxes with the height of the liquid is $$ q_{in}(t) - q_{out}(t) = A\dfrac{dh}{dt} \quad \text{,} $$ ...
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1answer
49 views

Theorem for showing the eigenvalues of a Jacobian matrix are less than one

After having the Jacobian matrix, I want to show whether the eigenvalues of this matrix are less than one or not. $ J=\left(\begin{array}{cccc} a & 0 & 0 & b \\ c & d & 0 & ...
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Lyapunov Stability for a Nonlinear, Time-varying system

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems. Say we are given a nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_2(t)[...
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1answer
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Properties of Positive Real Functions

I am trying to understand the properties of positive real (PR) and strictly positive real (SPR) transfer functions. If given a transfer function I know how to determine whether or not the function is ...
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37 views

Stability of a critical point for a general system ODE

Given an ODE-system I have proven that the solution $\textbf{x}(t)$ will have an upper bound of $\max\{|c_1|,|c_2|\}(\|\textbf{v}_1 \|+\| \textbf{v}_2\|)$. I then want to show that the solution is ...
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Dynamics $\delta x(t)=\delta x(0) e^{\lambda t}$ of Henon Attractor

Recall the question I asked before: Linearized perturbation dynamics of Henon Attractor So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$\delta x(t)=\delta x(0)...
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1answer
50 views

Linearized perturbation dynamics of Henon Attractor

I am reading the following paper: ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded) My question is around equation (2....
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51 views

Poincaré map under small pertubations

Let $\gamma$ be a periodic orbit of a vector field $X$. Through a point $x_{0} \in \gamma$ we consider a section $\Sigma$ transversal to the field $X$ and V a small neighbourhood which contains $x_{0}$...
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In stability analysis how to construct the jacobian matrix?

I'm a bit confused if we have $\dfrac{dx}{dt}=z+3y+x^2$ and $\dfrac{dy}{dt}=z^2$ what will be the components of the Jacobian matrix?
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What is the definition of a strong type?

I know the definition of a type over a set of parameters but can not find any definition for strong type. For example what does it mean to write $stp(a/A)$ ?