Questions tagged [stability-theory]
Stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.
494
questions
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13 views
Will a linear transformation of an ODE result preserve the convergence?
I have an ODE in $x$ defined over the strictly positive part of the simplex, meaning: $\sum_{j=1}^n x_j=1$ and that all $x_j$ are strictly positive.
I use the following linear transformation in order ...
0
votes
0answers
26 views
How can I prove the stability of this system? [closed]
Consider the following system:
$\dot{x}_1=x_2$
$\dot{x}_2=-x_2-sgn(x_1)$
where sgn(.) is the sign function. In simulations, both $x$ and its derivative converge to zero but I can not prove it ...
1
vote
0answers
24 views
Problems computing the Lyapunov spectrum depending on time interval T
I intent to determine the Lyapunov spectrum of nonlinear dynamic systems. To do so, I implemented some code in MATLAB, based on algorithms from literature and very similar to the way, described in ...
2
votes
0answers
38 views
Proof of Stability of parabolic-like Convolutional Residual Neural Networks
I am trying to understand a recent publication by Lars Ruthotto and Eldad Haber called Deep Neural Networks motivated by Partial Differential Equations published in the Journal of Mathematical Imaging ...
0
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0answers
27 views
Problem understanding bifurcations
I am beginning to study bifurcations, and I have some preoblems understanding some concepts.
I have understood that a Bifurcation can be defined has the change of behaviour of a dynamical system as a ...
0
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1answer
28 views
Determine stability of the equilibrium state
Please help me.
I am struggling to determine the stability of the equilibrium state $x = 0$ of the system
$$x_1' = x_1(x_1^2 + x_2^2 - \beta^2) + x_2 \\
x_2' = x_2(x_1^2 + x_2^2 - \beta^2) - x_1$$
...
0
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0answers
26 views
Proving stability where conservative vector field is given
while taking the vector calculus course, I had trouble solving the problem which is as follows:
Suppose a particle with mass $m$ moves in the space under the force field $- \nabla V $ where $V: \...
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0answers
8 views
Some questions in **Modulational stability of ground states of nonlinear Schrödinger equations**
Pictures below is from
Weinstein, Michael I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16, 472-491 (1985). ZBL0583.35028.
For $0<\sigma<\...
2
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1answer
36 views
Stability of equilibrium point origin for the equation $x'' + x + x^3(4 + \sin x) = 0$
The question asks to discuss the stability of equilibrium point origin for the equation $x'' + x + x^3(4 + \sin x) = 0$
Attempt: Given equation $x'' + x + x^3(4 + \sin x) = 0$
now taking $x' = y$ and $...
0
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1answer
49 views
How do I analyze the stability of a system with Lyapunov method?
I am a student and I am studying Lyapunov stability. I am considering the following system:
$\dot{x_1}=x_2+x_3$
$\dot{x_2}=-asinx_1-bx_2$
$\dot{x_3}=-asinx_1-x_3$
and I want to study its stability at ...
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0answers
31 views
How can I use Lyapunov Control for nonlinear system $\dot x = f (x, u) $
Recently I made my system identification algorithm SINDY to work. Not it can estimate a nonlinear model from measurement data that comes from a very nonlinear hydraulic system. The input signal ...
0
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1answer
32 views
Discrete-time Input-to-State Stability
From a famous control paper titled 'Input-to-state stability for discrete-time nonlinear systems', the following holds true: given a discrete-time system
$$x(k+1)=f(x(k),u(k))$$
let $V(x)=|x|^2$ where ...
1
vote
0answers
24 views
Help! Lyapunov proof for calculated torque control with friction term for robot
I want to prove asymptotic stability for a Calculated torque control with friction compensation. I was told to find "an already proved" system but I have had no luck while searching for books and ...
1
vote
1answer
79 views
Global Asymptotic Stability of a System
I have a system $V(x)$, in $R^2$, and I've calculated that
$V(x) \geq 0$ for all $x$ not equal to zero and that $V(0,0) = 0$
I've also calculated that $V'(x) \leq 0$
Since $V'(x)$ is NSD and not ...
0
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0answers
24 views
Finding the roots of a system of 4 differential equations
In Edward Routh's 1875 paper on the stability of Laplace's three body central configurations, the following system of differential equations is derived:
\begin{gather}
\begin{bmatrix}
b \mathrm{D}^{2}...
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0answers
54 views
Searching for unsolved problems in the field of stability
I have proposed an approach for constructing Lyapunov functions for autonomous systems in my Ph.D. thesis and find some useful examples. Now, I am searching for some another example in this field or ...
0
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1answer
19 views
Does linear discrete-time controllability imply stabilizability
Does linear discrete-time controllability imply stabilizability?
I feel like it should, since controllability is the ability to steer from any state $x(0)$ to another state $x(1)$ in finite time and ...
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0answers
37 views
Discover when zero solution is stable
Discover when $x=0, y=0$ is the stable solution of the system (depending on $a,b,k$)
\begin{cases}
\dot{x}=-y-x^k+ay^3\\
\dot{y}=x+bx^3-y^k
\end{cases}
$a,b\in\mathbb R, k\in\mathbb N$
...
0
votes
0answers
39 views
Stabilizability, what does it mean to steer to zero
The definition of stabilizability for linear systems is:
Stabilizability is the ability to steer a system to zero, with a control input that can be defined over an infinite amount of time.
This is ...
3
votes
0answers
57 views
Let $A\in\mathbb{R}^{2\times 2}$ and assume that $|{\rm tr}A|<4$. Prove or disprove that $b_1I_2+b_2A+b_3A^2+A^3=0$
Let $A\in\mathbb{R}^{2\times 2}$ and assume that $|{\rm tr}A|<4$. Prove or disprove that there exist $a_1,a_2,a_3,a_4\in\mathbb{C}$ such that $|a_1|<1$, $|a_2|<1$, $|a_3|<1$, $|a_4|<1$...
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0answers
86 views
Stability with input same as with nonlinear function?
Assume there is a dynamical system
$$
\frac{d x(t)}{dt} = A \cdot x(t) + q(x(t))
$$
and that $A$ is stable and that $q$ is a nonlinear and very complicated function. We only know $q$ is smooth and ...
0
votes
1answer
19 views
upper bound stepsize h
Consider the following initial value problem $$ y^{\prime} (t) = \lambda y (t) \, \text{for} \, t \in [0; T] ; y (0) = y_0; $$ with $T > 0, y_0 \in \mathbb{R} $ and $ \lambda \in \mathbb{C}$. ...
1
vote
0answers
35 views
Determining the domain of stability of a dynamic system
Suppose I have the system:
$\dot{x} = -x^3 - y^2$
$\dot{y} = xy - y^3$
... and am asked to find the domain of stability of the system.
Is my attempt and reasoning below deemed a correct approach?
...
0
votes
0answers
38 views
Relationship between Lyapunov functions and gradient Systems
given a nonlinear system
$f(z) = \dot{z}$
that induces gradient dynamics so that
$\nabla V(z) = -f(z)$
where V(z) is the potential function of the system. Is the potential function of a gradient ...
1
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0answers
18 views
Stability Analysis of Multi-Level PDE Difference Equation
I am working with the following problem:
$$v_t = \nu v_{xx}\quad v(x,0) = f(x)$$
I would like to analyze its stability using the discrete Fourier transform but have two questions:
How do you deal ...
1
vote
2answers
88 views
What is a suitable Lyapunov function for this system?
I have verified using the eigenvalue method that around $(0,0)$ the system
\begin{align}\dot x&=y - 3x - x^3 \\ \dot y &= 6x - 2y \end{align}
is stable. However, I have been trying to find ...
0
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1answer
19 views
Epidemology - interspecific competition, conditions for coexistence
Consider the inter-specific competition system
$$\frac{dx_1}{dt} = r_1 x_1 \left(1 - \frac{x_1 +\alpha_{12}x_2}{K_1} \right)$$
$$\frac{dx_2}{dt} = r_2 x_2 \left(1 - \frac{x_2 +\alpha_{21}x_1}{K_2}...
-1
votes
1answer
37 views
Bifurcation Near Origin of one parameter families of maps
I am working on a problem out of "An Introduction to Applied Nonlinear Dynamical Systems and Chaos" by S. Wiggins. Section 3 is all about local bifurcations and I am asked to describe the bifurcation ...
0
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0answers
24 views
MAPLE and equilibrium points
for example in this example
Solving system of nonlinear differential equation in MAPLE
I want evaluate the jacobian matrix of the system for each equilibrium point with mapel ?
1
vote
1answer
30 views
Definition of the Lyapunov exponents for compact operators
There is the following well-known result by Goldsheid and Margulis (see Proposition 1.3) on the existence of Lyapunov exponents:
Let $H$ be a $\mathbb R$-Hilbert space, $A_n\in\mathfrak L(H)$ be ...
1
vote
2answers
111 views
Problem understanding Center Manifold theory
I am studying stability for non linear control systems, and I am focusing on the Center manifold theory .
In particular, I am trying to understand an example which is also in the Hassan K.Khalil book ...
0
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1answer
28 views
Necessary stability condition for a second order discrete time system $x(k+2) = Ax(k+1) + Bx(k)$
Let $x(k) \in \mathbb{R}^n$, $A \in \mathbb{R}^{n\times n}$, $B \in \mathbb{R}^{n\times n}$. Consider the following discrete time system:
$$x(k+2) = Ax(k+1) + Bx(k)$$
where $x(1) = Ax(0)$ and $x(0) \...
1
vote
1answer
76 views
Problem understanding Chetaev theorem
I am studying control theory, and I am focusing on the Lyapunov stability. In particular, I am looking the Chetaev theorem, but I have some problems understanding it well.
I know that the Cheatev ...
0
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0answers
29 views
Stability of autonomous system $\frac{dy}{dx}=\cos(y)$
$$\frac{dy}{dx}=\cos(y)$$
Is critical point $y=\frac{3\pi}{2}$ semi-stable?
Because according to "Zill engineering math 6th", it says semi-stable means one side diverge to infinity and other side ...
0
votes
2answers
88 views
Why does the Lyapunov criterion only gives sufficient conditions for stability?
I am studying stability for control systems, and I have written in the notes of my professor that the Lyapunov Criterion only gives sufficient conditions for stability, and not necessary and ...
1
vote
1answer
38 views
Stability of linear time varying system
In the case of LTV systems, of the form
$\dot{x} = A(t)x$
the notion of uniformly globally asymptotical stability and globally exponential stability, are they one and the same? If possible, can ...
1
vote
1answer
64 views
Lyapunov exponents: Why do we know that the changes happen at an exponential rate
Let $E$ be a $\mathbb R$-Banach space, $\Omega\subseteq E$ be open, $f:\Omega\to\Omega$ be continuously Fréchet differentiable, $x_0\in\Omega$ and $\varepsilon>0$ with $B_\varepsilon(x_0)\subseteq\...
0
votes
1answer
45 views
Question on phase plane plot of ODE system
I am studying ODE systems myself and have an example in the book with the following ODE system\begin{equation*}
\begin{cases}
\dot x=x,
\\
\dot y = x+2y.
\end{cases}
\end{equation*}
The ...
0
votes
1answer
20 views
Prove that ODE solution is unstable using unstablitity definition
There is a ODE $\dot x=-2x, \dot y=3y$ . Prove using definition that at this point $x=0, y=0$ it is unstable.
I am not sure how to interpret the solution of this ODE and tell anything about lines ...
0
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0answers
23 views
Stability of second order linear systems
$\frac{dx}{dt} =-3x + 6y$
$\frac{dy}{dt} = 2x - 4y$
I found the Eigenvalues to be 0 and -7
The solutions says it’s a stable on line X=2y. How do I know this/ work this out?
0
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1answer
15 views
Linearization of control system
How can i linearize the following system ?
State equations:
$ x'_1 = (x_1-2)x_2 - 2x_2 $
$ x'_2 =-(x_1-2)^2 + x_2 + u -1 $
Output equation:
$ y = x_1 $
assume the eq. point : $ \tilde x=[2 ,0]^...
-1
votes
1answer
37 views
The stability of a fixed point, given that the one of the eigenvalues of the linearised system is zero and the other it negative?
I have the following dynamical system
$$\frac{d x}{d \tau}=\gamma x(1-x)-\alpha x y$$
$$\frac{d y}{d \tau}=y\left(1-\frac{y}{x}\right),$$
where $\gamma$ and $\alpha$ are constant parameters. I am ...
0
votes
0answers
25 views
To find class K infinity bounds on given radially unbounded function
Let $f = x_1^2 + x_2^4$. How to find class $\mathcal{K_{\infty}}$ functions $\alpha_1(||x||)$ and $\alpha_2(||x||)$, such that,
$\alpha_1(||x||) \leq f \leq \alpha_2(||x||), \forall x \in {R}^2.$
$\...
2
votes
1answer
26 views
Does stability along axes imply stability of the fixed point?
Let say I have a 2D-dynamical system
$$\dot x = f(x,y,\alpha_i)$$
$$\dot y = g(x,y,\alpha_i)$$
$i=1,2,...,n$ where $\alpha_i$ is a constant parameter.
Let $(x_0, y_0)$ be a fixed point, of which we ...
0
votes
1answer
90 views
Why for a linear system, the stability for a generic equlibrium point is equivalent to the stability of the origin?
I am studying the concept of stability for linear and for nonlinear systems. While studying the stability for a linear system I found this definition from the notes of my professor:
for a linear ...
0
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0answers
17 views
what is the geometric representation of Lyapunov stability
Take a look at the theorems below, in a dynamical systems linear or nonlinear, we construct a function usually the energy-function of a physical system, compute its derivative and apply the theorem to ...
1
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0answers
34 views
stability of a difference equation
I would like some help in investigating the stablity of the difference equation
$$
\begin{cases}
x_{n+1}=b x_n e^{ay_n} \\ y_{n+1}=b x_n (1-e^{-ay_n})
\end{cases}
$$ at (0,0).
I know that if b&...
3
votes
1answer
35 views
Defining equilibrium points of a 2D system with a variable as an equilibrium
Given the nonlinear system,
$$ x' = xy = f(x,y)$$
$$ y' = x(1-x) = g(x,y) $$
The two points where $f=g=0$ are $(0,y)$ and $(1,0)$. I am confused about the theory on how one would continue the ...
1
vote
0answers
27 views
Euler's Method stability for logistic equation
I have the ODE: $\frac{dP}{dt} = kP(1-\frac{P}{L})$ where $k$ and $L$ are constants.
I need to find the stability range for step size values for which Euler's method is stable for the above ODE. I am ...
0
votes
0answers
19 views
Convergence to stable points of damped equation of motion
Let's consider a function $U(x):\mathbf{R} \rightarrow \mathbf{R}$ (smooth "enough") with several local minima and the following damped equation for $x(t)$:
$$ \frac{d^2}{dt^2}x(t)=-\frac{d U(x)}{dx}-...
