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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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Find the right measure of stability for a very specific problem

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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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
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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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26 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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Mathematical correctness of dynamical system stability proof concept

If we have a system like this $$ \begin{align} \dot x_1=-a\sin(\arctan x_1)\\ \dot x_2=-b\sin(\arctan x_2)\\ \dot x_3=-c\sin(\arctan x_3)\\ \end{align} $$ where $a>0, b>0, c>0$. Obviously, ...
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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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26 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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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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2answers
56 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
34 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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1answer
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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
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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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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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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
39 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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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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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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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 ...
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How to determine the stability of the given polynomial?

Given a stable polynomial $\phi(s)=a_0+a_1{s}+a_2{s^2}+a_3{s^3}+\cdots+a_n{s^n}=\phi^{e}(s)+s\phi^{o}(s)$ where $\phi^{e}(s)=a_0+a_2{s^2}+a_4{s^4}+\cdots,~\phi^{o}(s)=a_1+a_3{s^2}+a_5{s^4}+\cdots$, ...
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Rate of convergence: Adaptive system

Consider the following dynamical system \begin{align} \dot{x}_1&=-ax_1 + w^T(t)x_2,\quad x_1\in\mathbb{R}^1 \\ \dot{x}_2 &= -w(t)x_1, \quad x_2\in\mathbb{R}^n \end{align} where $a>0$ and $w(...
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Stability of dynamical system via Lyapunov

I have a dynamical system which has the following form: $\dot x=\mathcal F_1(m_1)x+\mathcal F_2(m_2)x$. My objective is to find the parameters $m_1$ and $m_2$ via LMI (linear matrix inequality) using ...
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Why do Eigenvalues of Jacobian determine stability

I hear that positive eigenvalues of the Jacobian of the system imply the solution of the linearized system is of the form $e^{\lambda x}$ not $e^{-\lambda x}$ (where $\lambda$ is the eigenvalue), thus ...
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Stability of ODE involving trig functions and nonhyperbolic fixed points

Consider the following autonomous vector field: $$\dot x = −x$$ $$\dot y = \sin y$$ where $x \in \mathbb{R}^2, -\pi ≤ y ≤ \pi$ $\bullet$ Find all fixed points. $\bullet$ Determine the linearized ...
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1answer
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Stability of a 3D nonlinear ODE/dynamical system

I have tackled many 2D systems, but not any with 3D. I'm convinced that the principles and concepts still hold, however the problem is the lengthy computation of the eigenvalues as we have a 3x3 ...
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Hartman-Grobman theorem on a manifold

Consider a dynamical system $\Sigma$ on a manifold $M$ of dimension $d$ embedded in $\mathbb{R}^n$, where $d<n$. Let $x\in M$ be an equilibrium point and suppose we wish to determine the stability ...
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How to understand the Floquet Theory?

I am studying a physical case of a circular cylinder vibrating in a quiescent incompressible fluid. Such a system can be determined by two non-dimensional groups the Keulegan-Carpenter number and ...
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Center manifold of nonhyperbolic fixed point

Question: Consider the following autonomous vector field on the plane: $$\dot x = −x $$ $$\dot y = −x^2$$ $$(x,y) \in \mathbb{R}^2$$ $\bullet$ Compute the flow generated by this vector field. $\bullet$...
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How to describe a system with stability in terms of some variables

Assume I have a system: $\dot{\mathbf{x}}=-\frac{\partial}{\partial\mathbf{x}}L(\mathbf{x},\mathbf{y}) \\ \dot{\mathbf{y}}=\frac{\partial}{\partial\mathbf{y}}L(\mathbf{x},\mathbf{y})$ where $\...
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Conditions for which all orbits are periodic through varying constants.

Question: Consider the following autonomous vector field on the plane: $$\dot x = ax+by$$ $$\dot y = cx+dy $$ $$(x,y) \in \mathbb{R}^2 \qquad (a, b, c, d) \in \mathbb{R}$$ $\bullet$ Give a set of ...
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How do you obtain the fixed points and stability of a piecewise function?

Hi so I'm trying to work out how to find the fixed points of a piecewise function and do a stability analysis. One of the past exam papers in this complex systems course I'm doing has this question: ...
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If a time-varying discrete system matrix is bounded by two constant Schur stable matrices, what does it imply?

Consider a time-varying discrete system matrix $A_{t}$ that is bounded by two constant Schur stable matrices as follows \begin{eqnarray*} B & \leq A_{t}\leq & C, \end{eqnarray*} where the ...
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Why is stability of D.E. defined on infinite interval?

I will cite the following definition (from an O.D.E textbook) of the stability of a solution: Definition. Let $f:[0,\infty)\times\Omega\to\mathbb{R}^n$ be continuous and locally Lipschitz on $\Omega\...
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1answer
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Instability of a system subject to periodic perturbation (cont'd)

This is a follow-up to this question. Consider the following 2-dimensional system $$ \dot{x}(t) = A(t)x(t) \quad x(0)\in\mathbb{R}^2, $$ where $A(t)$ is a 2-dimensional time-varying matrix. Suppose ...
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Verifying stability of equilibria directly from the flow of an ODE

Question: Consider the following autonomous vector field on $\mathbb{R}$: $\dot x = x-x^3, x \in \mathbb{R}$ Compute all equilibria and determine their stability, i.e., are they Lyapunov stable, ...
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1answer
71 views

Instability of a system subject to periodic perturbation

Consider the following 2-dimensional system $$ \dot{x}(t) = A(t)x(t) \quad x(0)\in\mathbb{R}^2, $$ where $A(t)$ is a 2-dimensional time-varying matrix. Suppose that the origin of the above system is ...