Questions tagged [stability-theory]
Stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.
590
questions
0
votes
0answers
14 views
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.
0
votes
0answers
16 views
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 ...
0
votes
1answer
20 views
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|...
2
votes
0answers
31 views
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 ...
1
vote
0answers
25 views
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 $$(0,0) , (0,-2), (\frac{-2}{\sqrt3},0), (\frac{2}{\sqrt3},0)$$
...
0
votes
0answers
33 views
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 ...
0
votes
0answers
31 views
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)...
2
votes
0answers
36 views
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 ...
4
votes
3answers
78 views
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 ...
2
votes
1answer
60 views
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 ...
-1
votes
2answers
51 views
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 ...
1
vote
0answers
43 views
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. ...
0
votes
1answer
37 views
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 ...
1
vote
0answers
32 views
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?
$$w_t=-aw-b^2\int_o^{2\pi}...
1
vote
1answer
56 views
Stability of a degenerate equilibrium in a planer ODE using center manifold approach
I have a planer ODE system which is given by
\begin{eqnarray}
\begin{array}{lll}\begin{cases}
\frac{dx}{dt} = p_{20}x^2+p_{11}xy+p_{30}x^3+p_{21}x^2y+p_{40}x^4+p_{31}x^3y+p_{22}x^2y^2,\\
\frac{dy}{dt} ...
4
votes
2answers
92 views
Finding the region of attraction using a Lyapunov function
I'm trying to find an estimate for the region of attraction of an equilibrium point. The notes from Nonlinear Control by Khalil suggest that defining
$$
V(x) = x^TPx,
$$
where $P$ is the solution of
$$...
2
votes
0answers
27 views
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 ...
2
votes
2answers
110 views
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 ...
2
votes
0answers
35 views
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 ...
0
votes
0answers
22 views
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
$$\nu I\leq Q(t)\leq \rho I,$$
$$A^T(t)Q(t)+...
0
votes
0answers
12 views
Checking asimptotic stability of a critic point.
I've studied all the critical points of the following system using Hartman's Theorem, but for the (1,0) doesn`t apply because it's not hyperbolic. So I would like to find a strict Lyapunov function to ...
0
votes
1answer
23 views
Application on Gronwall's inequality
Using Gronwall' inequality, I need to show that the solution of the following initial value problem
$x'(t)=(1-a \cos{t})x$, $x(0)=x_0$
satisfies $|x(t)| \leq |x_0| e^{(1+a)t} $.
Here, $0<a<1$.
...
0
votes
0answers
10 views
About the truncation error of the scheme in the question
Question
I have been trying to solve this messy question for a while. I think, to solve this, Taylor series of the unknown terms must be expanded around point (x_m, t_n) such as this:
expansions
Is it ...
0
votes
0answers
11 views
For a SISO LTI system, does being internally stable imply BIBO stability?
For a single-input single-output LTI system,
$$ \dot x =Ax+Bu $$
$$ y =Cx $$
Does being internally stable, i.e. A is positive semidefinite, imply being BIBO stable?
3
votes
0answers
67 views
Stability of Explicit midpoint method
I am trying to determine the stability region of the well known explicit midpoint method
$$y_{i+1} = y_i + h f\left( t_i + \frac h 2, \ y_i + \frac h 2 f(t_i, y_i)\right)$$ and after following the ...
1
vote
0answers
18 views
Proving Stability of a Function of Laguerre Polynomials
I'm trying to prove that the following potential is stable at its critical point:
$$
F_{\textbf{n}}(x) = x - \sum_{\ell=1}^{r} \ln G_{n_{\ell}}(x),
$$
where $\textbf{n} = (n_1, n_2, \ldots, n_r)$ and
$...
0
votes
0answers
59 views
Stabilize a state of equilibrium
I'm having trouble solving a control theory problem.
I have a differential equation:
$$\dot{x}=-2x+Cx\gamma(t)$$
And I know that $|\gamma(t)|\le1$
Can I make the state of balance stable by choosing $C=...
1
vote
0answers
23 views
Stability of KdV equation
I want to find the stability of
$$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)\...
1
vote
0answers
24 views
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
$$
f(x)=
\begin{cases}
\mu x, & 0 \leq x\leq \frac{1}{2} \\
\mu - \mu x, & \frac{1}{2}\leq ...
0
votes
0answers
22 views
Convergence of monte-carlo methods.
Consider an algorithm to calculate $\pi$. Take a square $[0,1]\times [0,1]$ and make a quarter circle inside it of radius 1 centered at one of the corners. Now choose $N$ uniformly distributed random ...
2
votes
0answers
151 views
Stability of the equilibrium states
I have a function defined by the following differential equation
$$
\frac{\mathrm{d}\varphi}{\mathrm{d}t} = \gamma - F(\varphi)
$$
where
$F(\varphi)$ is a $2\pi$-periodic function) and
the chart of ...
0
votes
0answers
16 views
How to use Euler Theorem on homogeneous function to obtain $V' = -[2T - (\frac{\partial T}{\partial q}|q) - (\frac{\partial \pi}{\partial q}|q)]H$
Given the expression
$V' = -[-(q|\frac{\partial H}{\partial q}) + (\frac{\partial H}{\partial p}|p)]H$
where (a|b) denotes the scalar product, and H is the hamiltonian ($H = T(q, p) + \pi(q)$, and ...
1
vote
1answer
33 views
Stability theory and Lyapunov functions
Does Global Asymptotic Stability imply Global Uniform Asymptotic Stability? What conditions need to be satisifed for both types of stability?
1
vote
0answers
26 views
existence of a specific solution to a difference equation
I want to find the necessary conditions for positive sequence of real numbers, $\{a_n\}$ for which the following difference equation has a solution where $0 < \theta_n < \frac{\pi}{4}, \forall n$...
0
votes
0answers
13 views
Given $V=(q|p)H,\,H(q,p)=T(q,p)+\pi(q)$, what is $\frac{dV}{dt}$?
I am studying the book "Stability Theory by Lyapunov's Direct Method", by Rouche, Habbets and Laloy, and in theorem 2.10 it defines the function $V=-(q|p)H$, where $(q|p)$ is the scalar ...
1
vote
0answers
24 views
Routh Table First Column 0: Total number of RHP poles
I was doing some research and found that when a first column 0 appears but everything else is not necessarily 0 in the row, then there exists poles with nonnegative real parts (or positive real parts ...
0
votes
1answer
23 views
If $B(q)$ is a symmetric matrix and $B(0)$ is positive definite, is $T = p^T B(q) p$ positive definite?
In book "Stability Theory by Lyapunov's Direct Method" by Laloy there is a General Hypotheses concerning stability, which says:
"The kinetic energy is $T(q, p) = \frac{1}{2} p^TB(q)p$, ...
1
vote
0answers
38 views
Lyapunov stability f.o.differential equation
I have first order differential equation:
$$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}$$
How can I check the stability of the solutions $y(1)=0$...
1
vote
0answers
36 views
Check if this system has any attractor
We have the following system:
\begin{cases}
h'=h(1-h-a_{12}p) & \\
p'=\rho p(1-p-a_{21}h) &, \rho>0, a_{12}>1, a_{21}\in (0,1).\\
\end{cases}
We want to see if we have ...
3
votes
0answers
47 views
Unstable manifold of an unstable fixed point of a steepest descent iteration
Let $f:\mathbb R\to\mathbb R$ be differentiable $t:\mathbb R\to(0,\infty)$ be a "step size" function and $$\tau:\mathbb R\to\mathbb R\;,\;\;\;x\mapsto x-t(x)f'(x).$$ In the steepest descent ...
0
votes
0answers
24 views
Does there exist other version of Lyapunov-Like Lemma to solve this question?
Lyapunov-Like Lemma: If an energy function $\mathrm{V}(\mathrm{t}, \mathrm{x})$ satisfies the following conditions:
$\mathrm{V}(\mathrm{t}, \mathrm{x})$ is lower bounded
$\dot{\mathrm{V}}(\mathrm{t}, ...
1
vote
2answers
56 views
What is a suitable choice of Lyapunov function for this system?
I have the following nonlinear system:
\begin{align}\dot x&=x(2-y^4) \\ \dot y &= -3x(1+y^3) \end{align}
I first verified this system using linearization; however, the eigenvalues are $\lambda ...
0
votes
0answers
17 views
Asymptotic stability of compact sets
Let $\Omega$ be a topological space, $\tau:\Omega\to\Omega$, $A\subseteq\Omega$ and$^1$ $$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\...
2
votes
0answers
41 views
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$^1$ $$E(A):=\left\{x\in\...
0
votes
2answers
59 views
Relation between stability of fixed points of discrete and continuous dynamical systems
Let $f:\mathbb R^d\to\mathbb R^d$ be a diffeomorphism, $x_0$ be a fixed point of $$\frac{\rm d}{{\rm d}t}\varphi_t(x)=f\left(\varphi_t(x)\right)\tag1$$ (i.e. $f(x_0)=0$) and $A:={\rm D}f(x_0)$.
It is ...
1
vote
1answer
23 views
Does a global attractor contain every (forward) invariant set?
In Remark 2.1.1 of this work, it is claimed that "the global attractor of a dynamical system contains every invariant set".
It is not completely transparent what their definition of a "(...
0
votes
1answer
21 views
How can we show this set is stable, but not an attractor?
Let $\Omega$ be a topological space and $\tau:\Omega\to\Omega$. $A\subseteq\Omega$ is called stable if for every neighborhood $V$ of $A$, there is a neigheiborhood $U$ of $A$ with $$\tau^n(U)\subseteq ...
0
votes
1answer
27 views
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 $x$ ...
0
votes
0answers
20 views
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 $|\tau'(x^\ast)|<1$, ...
1
vote
0answers
20 views
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'...
