Questions tagged [stability-theory]

Stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.

Filter by
Sorted by
Tagged with
0 votes
0 answers
13 views

Identifying Bifurcation

I am trying to identify bifurcation of my $3D$ system near $a=6.58$. I am getting trajectories as shown in the picture and I am guessing it is Saddle-Node periodic orbit bifurcation since I am getting ...
SHR's user avatar
  • 1
1 vote
1 answer
63 views

Linearization of a nonlinear third order ODE and stability

I would like to know if the following differential equation ($\alpha,\beta,\gamma,d,\Lambda,w$ are constants) $x'''(t)=\frac{1}{24 (3 \alpha -\beta )}\frac{x(t)^{-3 w-2}}{x'(t)} \left(36 \alpha x(t)^{...
Axionlike particles's user avatar
0 votes
1 answer
59 views

Stability of discrete-time dynamical systems using Lyapunov stability where A is function of optimization variable

Hi I am trying to solve a constrained optimization problem using the Lyapunov stability. In the problem we aim to find $\beta$ such that $$\min_\beta ||\beta^TF-y|| \quad \text{s.t.}\quad A^{T}PA-P&...
geo200's user avatar
  • 1
1 vote
0 answers
25 views

Chetaev theorem for discrete time

In reading the following article: https://www.researchgate.net/publication/262736434_The_Chetaev_Theorem_for_Ordinary_Difference_Equations Theorem 1 seems to prove a discrete-time analog of Chetaev ...
xyz's user avatar
  • 940
1 vote
1 answer
63 views

Convergence with increasing Lyapunov function

Given a (autonomous) dynamical system, one can prove instability of a point via the Lyapunov method, by simply finding a Lyapunov function that increases in a neighbourhood of the point. This ensures ...
xyz's user avatar
  • 940
0 votes
0 answers
56 views

Singularity of a non- linear second order ODE

I have the encountered a singularity in the equation below . $$ y^{\prime \prime}(x)+\frac{2}{x} y^{\prime}+\left[y-\left(1+\frac{2}{x^2}\right)\right] y(x)=0, \quad 0<x<+\infty, $$ with ...
SR9054505's user avatar
2 votes
0 answers
21 views

Radially bounded Lyapunov function and global stability

I came accross this link about the necessity of the Lyapunov function being radially unbounded. My understanding is that this condition is unnecessary if the time derivative along solution ...
Yonatan's user avatar
  • 35
1 vote
1 answer
48 views

Lyapunov Stability class K functions

I'm reading the book on Nonlinear System Analysis by M. Vidyasagar. I see they define functions of class K as continuous strictly increasing functions such that $\phi(0)=0$ and from there, they define ...
user1880062's user avatar
0 votes
0 answers
66 views

Question on Lasalle's Invariance Principle

Consider system $\dot x = f(x)$ and let $\Omega\subset D$ be a positive invariant set. Let $V: D \rightarrow \mathbb R$ be a radially unbounded, positve definite function. Let the derivative fulfill $$...
Trb2's user avatar
  • 364
4 votes
1 answer
128 views

Exponential Stability and Lasalle's Invariance Theorem

It is well known that a system $\dot{x}=f(x)$ with $x \in \mathbb{R}^n$ is exponentially stable if there exists a Lyapunov function $V(x)$ which satisfies \begin{align} k_1\Vert x \Vert \leq V(x) &...
Trb2's user avatar
  • 364
2 votes
1 answer
58 views

In control systems, What inputs will make system state unbounded while system output bounded?

Suppose I have a discrete LTI state space system: $$x(t+1)=Ax(t)+Bu(t),$$ $$y(t)=Cx(t)+Du(t),$$ What i am courious is that under what conditions the input $u(t)$ satisfies would make $lim_{t\...
JambooRee's user avatar
0 votes
0 answers
28 views

Checking Positivity

enter image description here I am trying to check positivity of an algebraic expressions where all the variables are positive. Using Mathematica 9.1 version I have got the result showed as follows. ...
SHR's user avatar
  • 1
2 votes
2 answers
74 views

Which stability criterion to use for LTI system with Gaussian noise?

This might (and hopefully will) be a very simple question but I'm quite stumped after doing some research: consider the noise-free discrete-time LTI dynamics given by $ \boldsymbol{x}(k+1) = \...
Bart Wolleswinkel's user avatar
1 vote
0 answers
69 views

Khalil's proof of local Lyapunov stability incomplete?

This is regarding the proof of a central theorem (Theorem 4.1) in Hassan Khalil's seminal Nonlinear systems book: Theorem 4.1: Let $x = 0$ be an equilibrium point for $\dot{x} = f(x)$ where $f:D\...
user1814274's user avatar
0 votes
1 answer
35 views

Lyapunov matrix equation theorem

I know the following Lyapunov's theorem: For any symmetric positive definite matrix $Q$, the following are equivalent 1- The matrix equation $A^T X + X A = -Q $ has a unique solution $X$ that is ...
Mathisfreedom's user avatar
0 votes
0 answers
39 views

from local stability to global stability

Suppose I have the system $x'=F(x)$ with $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$. I denote by $J(x)$ the Jacobian matrix, that is, $J_{ij}(x)=\partial F_i/\partial x_j (x)$. Suppose I know that for ...
Yonatan's user avatar
  • 35
0 votes
2 answers
68 views

Question about Cetaev Theorem

Cetaev Theorem states that: "Considering the ode $X'=F(X)$, with an equilibrium $x_{0}$. If there is a function $V$: $U_{0} \rightarrow $ IR and a region $\omega$ in $U_{0}$ that contains $x_{0}$,...
Albi's user avatar
  • 29
1 vote
0 answers
58 views

A delayed differential equation system and its characteristic polynomial

I have the following DDE system: \begin{equation} \begin{split} \dot{x}_{1} &= -\mu x_{1}(t) + a_{11}f_{11}(\int_{-\infty}^{t} F(t-s) x_{1}(s-\tau_{1})ds) + a_{12} f_{12}(x_{2}(t-\tau_{1}))) \\...
neuralode's user avatar
1 vote
0 answers
48 views

Find a stable numerical solution of differential equation $y''=g(x)y$

I'm attempting to numerically solve this differential equation using MATLAB's ode45, presuming a stable solution, i.e., $y$ remains finite as $x\to+\infty$. ($g(x)$ is a pending function determined by ...
Knifer Plasma's user avatar
1 vote
1 answer
53 views

Numerical stability of spherical Bessel recurrence relation

I'm working on a homemade code that needs an implementation of spherical Bessel function of the first kind myself. The order is restricted to non-negative integers, and only real argument is required. ...
jinzx10's user avatar
  • 109
1 vote
0 answers
36 views

How is this stability bound for the unique interpolant possible? $|s^*(x)|^2\le K(x,x)\|f\|_K\text{cond}_2(G_S)$

If one wants to reconstruct a function $f$ which, we assume is an element of a Hilbertspace $(\mathcal{H}(\Omega,K),(\cdot,\cdot)_K)$ of functions $\Omega\to\mathbb{R}$ with a reproducing Kernel (a.k....
Max Stuthmann's user avatar
1 vote
1 answer
70 views

Estimate solution of nonlinear differential equation by linear differential inequality [closed]

Consider the linear ordinary differential equation: \begin{align*} \dot x(t) & = Ax(t), \\ x(0) & = x_0, \end{align*} $t\geq 0$, $x\in\mathbb{R}^n$, with $A$ is Hurwitz (i.e., all its ...
MathsStudent's user avatar
0 votes
0 answers
55 views

How do I solve this problem with Lyapunov functions?

Hi this is the system i need to work on: $$ \left\{\begin{array}{rcl} \dot x&=&x(1-x)-axy\\ \dot y&=&y(1-y)-bxy \end{array} \right. $$ I found four stationary points: $(0,0)$;$(1,0)$;$(...
Marzio's user avatar
  • 1
1 vote
1 answer
63 views

Well-Posed ODEs

I'm somewhat troubled by how the notion of a well-posed problem extends to ordinary differential equations. It is commonly said that a problem is well-posed if following three criteria are met:- The ...
zaccandels's user avatar
0 votes
1 answer
52 views

Find a Lyapunov function for this nonlinear system

This was a problem that I encountered about a year ago: Use the Lyapunov function method to determine the stability of the equilibrium of the origin of this system: $$(x_1)' = -x_1 + x_2 - x_{1}^{3}$$ ...
random's user avatar
  • 71
2 votes
2 answers
92 views

Determine stability of non-hyperbolic stationary point

Given the system $$\begin{align*} \dot{x_1} &= x_2+x_1^2-x_1^3 \\ \dot{x_2} &= -x_2+\mu x_1^2 \end{align*} $$ determine the stability of the stationary point in the origin for $\mu = \{-1,0, 1\...
Carl's user avatar
  • 519
0 votes
0 answers
16 views

LaSalle's invariance principle for positive semi definite V

With $\dot{\textbf{x}}=\dot{\textbf{x}}(\textbf{x})$, if I have a function which has $V(\textbf{x})>0$ when $\textbf{x}\neq \textbf{0}$ for some domain of x, and if $\dot{V}(\textbf{x})\leq 0$ for ...
Minecraft dirt block's user avatar
0 votes
1 answer
69 views

Bifurcation Analysis of Non-autonomous system

Suppose we have a matrix equation $$ \frac{d}{dx}\mathbf{u} = \mathbf{A}(x) \mathbf{u} + \mathbf{b}. $$ If $\mathbf{A}(x)=\mathbf{A}$ were constant, then one can inspect the eigenvalues of the ...
AQuestion's user avatar
1 vote
0 answers
35 views

With $A=A^T$, $Ax=b$ is solved backward stably for $x$ yielding computed $y$. Show that $y / | y |$ is close to $x / | x |$.

Suppose $A \in \mathbb{R}^{m \times m}$ is a real symmetric matrix with orthonormal eigenvector basis $\{ \vec{q}_1, \ldots, \vec{q}_m \}$. Let $b \in \mathbb{R}^m$ be expressed in this basis as: \...
clay's user avatar
  • 2,687
0 votes
0 answers
42 views

Stability of MIMO system by system matrix eigenvalues

I have a MIMO LTI system described by $\begin{cases}\mathbf{\dot{x}=Ax+Bu}\\ \mathbf{y=Cx} \end{cases}$ If it is controllable, can I conclude system's BIBO stability provided that all eigenvalues of $...
Mathisfreedom's user avatar
0 votes
1 answer
25 views

Can we derive an exponential bound from a special limit of a function?

Assume $\lambda>0$ and a continous function $x(t), x:[0,\infty) \rightarrow \mathbb{R}^n$. Does it hold that $$ \lim_{t \to \infty}\frac{1}{t} \log(|x(t)|)<0 \Leftrightarrow \exists C>0: \...
Keine_Maschine's user avatar
0 votes
0 answers
38 views

The real CFL condition for cylindrical laplacian

I've been exploring the CFL (Courant-Friedrichs-Lewy) condition in polar coordinates and have observed that previous inquiries haven't yielded a satisfactory answer. I've come across this paper which ...
Manuel Borra's user avatar
0 votes
0 answers
11 views

How do I fix this transfer function with algebra?

Currently, I'm working on a battery analysis system. For now, I have this transfer function: $$\frac{V_L(z)}{I_L(z)} = \frac{\left(\frac{V_{oc}}{I_L(z)}\right) +b_0 - b_1z^{-1}}{a_0 - a_1z^{-1}}$$ ...
Thor-x86_128's user avatar
0 votes
0 answers
45 views

Stability of normal state in chemostat model

The chemostat model proposed by monod was given by, $$ \begin{align} \frac{dx}{dt}&=[K(c)-D]x\\ \frac{dc}{dt}&=D[c_0-c]-\frac1yK(c)x \end{align} $$ where $x(t)$ is the population of micro-...
N00BMaster's user avatar
0 votes
0 answers
16 views

Negative semi-definiteness stability

I am working on the stability of a system for the estimation $z_i$ predicted from x using the basic observer design. The relation is as follows: \begin{equation} z_i = \frac{\eta_i }{k_i}(\dot{x} - \...
KASSIM S. O.'s user avatar
3 votes
1 answer
122 views

Stability and Asymptotical behavior of a nonlinear system

A simple mathematical model to describe how the HIV/AIDS virus infects healthy cells is given by the following equations: $$ \begin{align} \frac{dT}{dt} &= s - dT - \beta Tv \\ \frac{dT^*}{dt} &...
N00BMaster's user avatar
0 votes
0 answers
31 views

Stability of Forward Euler in nonlinear ODE

I have the following ode: $y^{'}= \frac{k}{\sqrt{y}}$ where k is a positive value. Applying the Forward Euler method gives the following: $v^{n}=\frac{\Delta t \ k}{\sqrt{v^{n-1}}}+v^{n-1}$ I'm ...
Camilo Andrés Acevedo Ardila's user avatar
3 votes
1 answer
47 views

Proving that the zero solution of a linear periodic ODE system is unstable

Prove that the zero solution is unstable for the system $x' = A(t)x$ with $A = \begin{pmatrix} \frac{1}{2} - \cos(t) & 12 \\ 147 & \frac{3}{2} + \sin(t)\end{pmatrix}$. I've tried the following:...
arinarmo's user avatar
  • 141
0 votes
0 answers
29 views

Stability by the Von Neumann criterion

I've just determined the explicit numerical method to solve the burgers equation $$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = v \frac{\partial^2 u}{\partial x^2}$$ in one ...
dabib's user avatar
  • 11
0 votes
0 answers
27 views

Is it reasonable to evaluate the monodromy matrix of a periodic solution numerically with limits?

Suppose we are evaluating the stability of a periodic solution of a dynamic system: $$ \dot{x}=f(x,t) $$ where $x\in \mathbb{R}^n$, and $f$ being a smooth vector field periodic in $t$ with $f(x,t) = f(...
Gape's user avatar
  • 11
0 votes
0 answers
46 views

The relationship between real negative eigenvalues and convergence rate for ODE.

Let $\pmb{\delta}=\pmb{\delta}^\triangle$ be an equilibrium point for the following ODE , \begin{align*} \frac{\partial \pmb{\delta}(t)}{\partial t}=\pmb{F}(\pmb{\delta}) \ with \ \ \pmb{F}(\...
Fight for ambition's user avatar
2 votes
1 answer
59 views

What is the probability that a matrix with i.i.d. normal entries is stable?

Let $A$ be an $n \times n$ random matrix, such that the entries $a_{ij}$ are i.i.d. from the standard normal distribution. I'm curious on the probability that $A$ is Schur stable. That is, $$P(\rho(A) ...
Spencer Kraisler's user avatar
0 votes
0 answers
61 views

limit cycle in high dimension system (8 dimension for example)

I am very new to the nonlinear dynamical-systems. I wonder if there is a method to determine when a limit cycle situation will occur in the non-linear dynamical system, which is described by the ...
jhdai's user avatar
  • 1
3 votes
0 answers
69 views

How bistable is my system?

Description: given ODE: $\dot{x} = a + bx +cx^2 +dx^3$, I have mutliple combinations of the coefficients $a,b,c,d$ that I want to understand whether they make a bistable or not system. For this ...
athantas's user avatar
1 vote
1 answer
41 views

Robotic Control and Parameter Uncertainty: The Significance of Incremental Stability Analysis

Question: In the field of control theory and robotics, incremental stability is a concept that extends our understanding of system behavior. Consider a practical example involving a robotic ...
YAKINDA's user avatar
  • 55
0 votes
0 answers
71 views

Why is my numerical scheme for heat equation unstable?

I've got an inhomogeneous heat equation at hand: $$\partial_tu=\alpha\partial_x^2u+f(t,x).$$ I discretised it according to the FTCS scheme, i. e. I applied the forward Euler method for $\partial_t$ ...
thorr's user avatar
  • 23
1 vote
0 answers
42 views

Meaning of complex eigenvalues for 2D matrix realtive to dynamical systems

I am studying non-linear dynamical systems with the linearization method around an equilibrium point, but I don't get the geometrical meaning of complex eigenvalues. (Let's focus on a 2D case) For ...
ohhConti's user avatar
2 votes
1 answer
134 views

What if the level set of Lyapunov function is disconnected? - when estimating region of attration

Consider $\frac{dx}{dt}=f(x)$, where $x\in\mathbb{R}^n$. Suppose $x=0$ is a stable equilibrium. It is classical way to estimate region of attraction of $0$ by finding a $C^1$ function $V(x)$ such that ...
happyle's user avatar
  • 183
1 vote
0 answers
51 views

Lasalle's invariance principle for global stability of synchronization state of Kuramoto model

question My question regarding the argument of the proof of theorem 3.1 in this paper. In the proof the Lasalle's invariance principle is used. From what I learned, radially unboundedness must hold ...
happyle's user avatar
  • 183
-2 votes
1 answer
66 views

Are there any concrete application of the Lyapunov theorem for LTI systems?

Consider a LTI system $\dot x = Ax$. This system is globally asymptotical stable iff given any $Q \succ 0$, there exists a unique $P \succ 0$ such that $A^{T}P+PA+Q=0$ holds. https://en.wikipedia.org/...
Fraïssé's user avatar
  • 11.2k

1
2 3 4 5
17