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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $$x_1' = x_1(x_1^2 + x_2^2 - \beta^2) + x_2 \\ x_2' = x_2(x_1^2 + x_2^2 - \beta^2) - x_1$$ ...
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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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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<\...
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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 $...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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} $a,b\in\mathbb R, k\in\mathbb N$ ...
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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 ...
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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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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 ...
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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}$. ...
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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? ...
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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 ...
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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 ...
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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 ...
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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}...
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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 ...
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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 ?
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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 ...
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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 ...
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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) \...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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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 ...
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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?
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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]^...
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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 ...
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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, $\alpha_1(||x||) \leq f \leq \alpha_2(||x||), \forall x \in {R}^2.$ $\...
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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 ...
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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 ...
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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 ...
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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&...
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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 ...
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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 ...
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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}-...

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