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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0answers
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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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0answers
15 views
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 ...
3
votes
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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0answers
62 views
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 ...
2
votes
0answers
19 views
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 ...
0
votes
0answers
31 views
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(...
0
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1answer
22 views
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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0answers
20 views
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 ...
0
votes
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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0answers
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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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0answers
39 views
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 $\rho:\mathbb{R}\to\mathbb{R}$ ...
3
votes
0answers
48 views
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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0answers
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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0answers
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
$$x'_i=-\sum_{j=1}^n[a_{ij}\,d_{1j}(t)+b_{ij}\,d_{...
1
vote
1answer
36 views
Stability of non-hyperbolic critical point in two dimensions, where the linearization has one zero egenvalue
I am having a two dimensional autonomous system
$S' = 2S^3+2S^2+\frac{1}{2}SA-\frac{3}{2}S-\frac{3}{4}A$
$A' = 4AS^2+A^2+4AS$
which exhibits a critical point at the origin $(S,A)=(0,0)$, and others....
1
vote
0answers
26 views
Stability of equilibrium points of system of differential equations
Given $$\ddot{x}+\lambda\dot{x}=x-x^3,$$
which I rewrote to
\begin{align}
\dot{x}&=y \\
\dot{y} &= -\lambda y + x - x^3,
\end{align}
For the exercise we take $\lambda\geq 0$.
I have determined ...
1
vote
0answers
40 views
Creating equilibria by adding a time-periodic term to an ODE?
Let $f\colon \Bbb R^N\to \Bbb R^N$, $g\colon[0,\infty)\to\Bbb R^N$ and consider the ODE
$$ \dot x(t) = f(x(t)) + g(t), $$
where space- and time-dependence are additively separated.
Is it possible to ...
1
vote
0answers
20 views
Complex dynamical system: check stability of point numerically
Given a dynamical system described by the equations (for i=1,...,N)
$$\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$$
Say that I have ...
0
votes
0answers
29 views
Direction Field in MatLab
I need to plot the direction field and phase plane of the following ODE using matlab. I've tried using meshgrid w/ the quiver function, however, I'm not getting the correct field.
I need to plot
$\...
1
vote
1answer
29 views
Finite difference stability problem
My apologies for the title I'm not quite sure how to title a problem like this.
I need to show the following result:
$$u_j^{n+1} = e^{\Delta t\partial/\partial t}u_j^n$$
Where $u_j^n$ is the ...
4
votes
1answer
68 views
Estimate in proving that linear instability implies nonlinear instability
I am trying to prove the exercise on page 3 of http://depts.washington.edu/bdecon/workshop2012/g_stability.pdf. This question was already asked before here: Linear instability implies nonlinear ...
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votes
0answers
18 views
Eigenvalues of a block matrix and stability of a linear system of differential equations with time varying coefficients
Let $X \in \mathbb{R}^{n \times 1}$ and $P_1,P_0(t) \in \mathbb{R}^{n \times n}$ symmetric and positive definite for all $t \in \mathbb{R}$. Consider the dynamical system:
$$ \ddot{X} + P_1\cdot \dot{...
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votes
0answers
10 views
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
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}<\...
2
votes
0answers
43 views
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 ...
0
votes
0answers
14 views
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 ...
2
votes
1answer
22 views
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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0answers
48 views
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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votes
1answer
23 views
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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0answers
29 views
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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votes
0answers
29 views
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 ...
1
vote
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 ...
0
votes
2answers
39 views
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\...
0
votes
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 &...
2
votes
2answers
53 views
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 ...
0
votes
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 ...
0
votes
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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votes
0answers
42 views
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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votes
0answers
24 views
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
$$
q_{in}(t) - q_{out}(t) = A\dfrac{dh}{dt} \quad \text{,}
$$
...
1
vote
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 & ...
3
votes
0answers
68 views
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:
$$\dot{x_1}(t)=-x_1(t) + x_2(t)[...
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1answer
91 views
Properties of Positive Real Functions
I am trying to understand the properties of positive real (PR) and strictly positive real (SPR) transfer functions. If given a transfer function I know how to determine whether or not the function is ...
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0answers
37 views
Stability of a critical point for a general system ODE
Given an ODE-system I have proven that the solution $\textbf{x}(t)$ will have an upper bound of $\max\{|c_1|,|c_2|\}(\|\textbf{v}_1 \|+\| \textbf{v}_2\|)$. I then want to show that the solution is ...
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votes
0answers
35 views
Dynamics $\delta x(t)=\delta x(0) e^{\lambda t}$ of Henon Attractor
Recall the question I asked before:
Linearized perturbation dynamics of Henon Attractor
So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$\delta x(t)=\delta x(0)...
1
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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....
1
vote
0answers
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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0answers
37 views
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?
2
votes
0answers
35 views
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 $stp(a/A)$ ?
