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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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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 ...
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1answer
43 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 ...
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18 views

Behavior of the iterated map $x_n=C-1/x_{n-1}$ when $|C|\leq 2$

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\...
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28 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 &...
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41 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 ...
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1answer
15 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 ...
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21 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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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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21 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 $$ q_{in}(t) - q_{out}(t) = A\dfrac{dh}{dt} \quad \text{,} $$ ...
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1answer
42 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 & ...
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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
59 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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32 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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33 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)...
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1answer
46 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....
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49 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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28 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?
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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 $stp(a/A)$ ?
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171 views

Measure of stability

I am working on a machine learning project when I realized I add a question. This is not programming, nor statistic, nor a probability question, but a real pure mathematical question. So I think my ...
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35 views

How to draw the phase portrait if two of the eigenvectors are (0,0,0,0) and eigenvalues are positive in one case and negative in other?

I have two eigenvalues and they are 2wsqrt(2) and -2wsqrt(2) where w is the angular frequency of the system. Both of the eigenvectors corresponding to these eigenvalues are (0,0,0,0) so how can one ...
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1answer
25 views

determine the stability of an equilibrium point(x,0).

I would like to determine the stability of equilibrium point $(x,0)$ of the differential equation $\dot x = Ax$ $ A= \bigg[ \begin{matrix} 0&0\\0&a \end{matrix} \bigg] $ and $ a >0 $ ...
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1answer
38 views

How to rotate a coordinate system to find the unstable manifold.

I am considering the dynamical system: $u'=v-0.25(v-u)^2$ $v'=u(1+v)+0.25(u+v)^2$ I have calculated the linear stable and unstable manifold as, $E^s=sp(1,1)$ and $E^u=sp(-1,1)$ for eigenvalues $-1$ ...
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Determine stability of non autonomous system at the origin

I am trying to determine the stability of the zero solution of the system $x'= \begin{bmatrix} -t & 1 \\ 1 & -t \end{bmatrix}x $ Even though, a Liapunov method can only ...
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BIBO Stability in Z-domain

I'd really appreciate it if someone could please explain to me the condition for a LTI system to be BIBO stable, in z-domain. I have a background in control, and in linear control for example, if we ...
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Is there any theoretical result on how to stabilize a polynomial by changing its coefficients?

The stability of a general $n$ order polynomial is associated with the following statement: if all the roots of the following equation falls in unit circle on the complex plane, then the system is ...
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Convergence to a fixed value

This question is a follow up on the question about marginal stability of LTI system: \begin{equation} \mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1] \end{equation} I am interested in the algorithm: $k :=...
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45 views

How to find the equilibrium points of this dynamical system?

Consider the dynamical system $$\dot x= cx - \frac{x}{1+x^{2}}$$ for $x\in\mathbb{R}$, with $c$ a positive constant. Establish the location and number of equilibrium points of the system for all ...
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1answer
38 views

Solving system of differential equations, checking for stability and plotting the result

I am trying to solve the following system, but I am not sure if I am doing it properly \begin{equation} \mathbf{\dot{y}} = \mathbf{Ay} \text{ where } \mathbf{A} = \begin{bmatrix} -2 & 1 \\ -1 ...
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Stability of the steady state of a linear transport PDE

I am working with the PDE for $x>0$ and $t>0$ : $$\frac{\partial}{\partial t } n(x,t)+\frac{\partial}{\partial x} g(x) n(x,t)=-\mu(x)n(x,t) $$ where the characteristic of the problem is: $$ \...
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92 views

How to pick a Lyapunov function and prove stability?

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for autonomous systems. Say we are given the nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_1(t)x_2(t)...
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1answer
48 views

Marginal stability of discrete linear time-invariant system

I have a question about marginal stability of a system: \begin{equation} \mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1] \end{equation} I would adapt the definition of marginal stability from this ...
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1answer
24 views

Linear stability analysis on a simple pendulum

So I have a simple pendulum (rod has no weight, point mass, no frictional forces) and I’m measuring the angle theta from the downward vertical, hence I have the governing equation $$\ddot{\theta}+sin(\...
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81 views

Implication of stability of Van der Pol oscillator.

Consider the homogeneous Van der Pol equation, $\ddot{x} + \mu (x^2-1)\dot{x} + x = 0$, with $\mu>0$. We convert it into a dynamical system, $$\dot{\bf x} = (y, -(x+\mu(x^2-1)y), \ \mathbf{x} \...
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1answer
23 views

Using PI control to eliminate steady state errors

In a negative feedback loop i understand the mistake of canceling unstable poles. But take for example a plant $G(s)=\frac{1}{s+1}$ and an I-control $F(s)= \frac{1}{s}$ Then the system has the ...
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Internal stability of a discrete-time system

These are two parts of a much larger proof I'm working on, can't figure how ii implies iii though. $x(k+1)=Ax_k,x(0)=x_{0}$ Where $A∈\mathbb{R}^{n×n}$ is a real constant matrix. i) All the ...
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Discrete-Time External Stability

Consider a discrete-time system $\sum_{L}^{}$ of the form $x(k+1) = Ax(k) + Bu(k)$ $y(k) = Cx(k)$ Show that if all the eigenvalues of A are on the open unit disc, show that $\sum_{L}^{}$ is BIBO ...
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2answers
52 views

Determining Bifurcation of a Function

Hello I am trying to find analyze the bifurcation behavior of $\dot{N} = N(N - e^{\alpha N}) , N \geq 0, \alpha > 0$ as $\alpha$ is varied and find their stability. Playing around with different ...
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How to design a Robust observer for a 2D system

Consider the second order system given by $\dot{x}=Ax+Bw(t)$, where $x\in\mathbb{R}^2$, $$A = \begin{bmatrix} {0},{6}\\ {-1} {-6} \end{bmatrix}, \quad B = \begin{bmatrix} {0}\\{1}\end{bmatrix}...
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2answers
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Showing that a centre of the 2D linear system $\dot{\mathbf{x}} = A \mathbf x$ is Lyapunov stable

Consider the 2D linear system $\dot{\mathbf{x}} = A \mathbf x$ with $$A = \begin{pmatrix} 0&1\\ -4 & 0\end{pmatrix}.$$ The eigenvalues of this matrix are $\lambda = \pm 2i$, meaning that the ...
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1answer
99 views

What's an example where Lyapunov fails to find the bounds of stability

In linear control theory, a system is stable if and only if if satisfies the Routh–Hurwitz stability criterion, so we can use this to solve for the limits of stability. E.g. you can find the maximum ...
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Does differentiating an integro-differential equation results in equivalent stability of the solution?

Consider the following integro-differential equation: $$\dot{x}(t)=ax(t)+b\int_0^tx(\tau)\text{d}\tau,$$ where $\dot{x}(t)$ denotes the time derivative of $x(t)$. If we derive the above equation and ...
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1answer
28 views

Differential equations, convergence?

I am dealing with the following matrix. $A=\begin{pmatrix} 0&a & a & a & c & c &c & c\\ a& 0& a &a & c& c& c& c\\ a& a &0 &...
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1answer
43 views

Reason for choice of the word “asymptotically” stable in Lyapunov stability theory?

The equilibrium $x^\ast$ is (Lyapunov) stable iff $$\forall \varepsilon > 0 \; \exists \delta(\varepsilon) : \lvert x(0) - x^\ast \rvert < \delta(\varepsilon) \Rightarrow \forall t \geqslant 0 \;...
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Harvesting equation bifurcation

Could anyone give me some pointers on how to make a bifurcation diagram of a two parameter ODE of a harvesting model. It's $$ \dot{x} = ax\left(1 - \frac{x}{b}\right)- \frac{x^2}{1 + x^2}. $$ If I ...
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1answer
106 views

First Integral of Pendulum with Friction

How can we prove that an ODE does not have a first integral (i.e., a constant of motion that is conserved along the trajectories)? For example, is it true that the pendulum system of ODEs $$\dot x_1 = ...
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How to interpret Jacobi Stability?

I'm having the first contact with Jacobi stability for second order ODE, and I didn't understand very well what is the difference between the concept of Lyapunov stability nd Jacobi. It's very well ...
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If $|\lambda| < 1$ for every eigenvalue of $A$ in $x_{i+1} = Ax_{i}$ then $0$ is an asymptotically stable fixed point.

Consider the map $x_{i+1} = f(x_{i})$ and let $x^{*}$ be a fixed point then the fixed point is said to be Lyapunov stable if $\forall \epsilon > 0 $ there exist $\delta > 0$ such that $x_{i} \in ...
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50 views

A question about stability

A dynamical system is governed by the equation $\frac{dx}{dt}=2\sqrt{1-x^2}$, $|x|\leq 1$. Then By equating $\frac{dx}{dt}$ to $0$ we get $1,-1$ are the fixed points. But how to check their stability? ...
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1answer
58 views

Disprove/Prove Existence of Periodic Solution for Autonomous ODE

Consider the system $\dot x = x^2 + y^2 -1$ and $\dot y = y - 2xy$. I am new in this field. I draw the vector filed and I saw that there is no obvious periodic solution. How can I prove/disprove the ...
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1answer
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Stability proof of nominal MPC with terminal cost and constraint

While going trough these slides, I wasn't able to make sense of the following on slide 32: (if only providing the url to the slides is not ok I will edit the question, but doing it like that saves a ...