Questions tagged [stability-theory]
Stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.
0
votes
0answers
8 views
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 ...
0
votes
0answers
16 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 ...
1
vote
1answer
18 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 $
...
0
votes
1answer
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 $-1$ ...
1
vote
0answers
45 views
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, ...
-1
votes
2answers
53 views
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 ...
1
vote
0answers
15 views
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 ...
0
votes
0answers
10 views
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 ...
0
votes
0answers
20 views
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 :=...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
0answers
16 views
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:
$$ \...
3
votes
2answers
56 views
How to pick a Lyapunov function and prove stability?
I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for autonomous systems.
Say we are given the nonlinear system:
$$\dot{x_1}(t)=-x_1(t) + x_1(t)x_2(t)...
0
votes
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 ...
0
votes
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 $$\ddot{\theta}+sin(\...
1
vote
1answer
64 views
Implication of stability of Van der Pol oscillator.
Consider the homogeneous Van der Pol equation, $\ddot{x} + \mu (x^2-1)\dot{x} + x = 0$, with $\mu>0$. We convert it into a dynamical system, $$\dot{\bf x} = (y, -(x+\mu(x^2-1)y), \ \mathbf{x} \...
0
votes
1answer
18 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 ...
0
votes
0answers
14 views
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 ...
1
vote
0answers
20 views
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 ...
3
votes
2answers
46 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 ...
0
votes
0answers
20 views
How to design a Robust observer for a 2D system
Consider the second order system given by $\dot{x}=Ax+Bw(t)$, where $x\in\mathbb{R}^2$,
$$A = \begin{bmatrix}
{0},{6}\\
{-1} {-6}
\end{bmatrix}, \quad B = \begin{bmatrix} {0}\\{1}\end{bmatrix}...
1
vote
2answers
35 views
Showing that a centre of the 2D linear system $\dot{\mathbf{x}} = A \mathbf x$ is Lyapunov stable
Consider the 2D linear system $\dot{\mathbf{x}} = A \mathbf x$ with $$A = \begin{pmatrix} 0&1\\ -4 & 0\end{pmatrix}.$$ The eigenvalues of this matrix are $\lambda = \pm 2i$, meaning that the ...
4
votes
1answer
95 views
What's an example where Lyapunov fails to find the bounds of stability
In linear control theory, a system is stable if and only if if satisfies the Routh–Hurwitz stability criterion, so we can use this to solve for the limits of stability. E.g. you can find the maximum ...
1
vote
0answers
29 views
Does differentiating an integro-differential equation results in equivalent stability of the solution?
Consider the following integro-differential equation:
$$\dot{x}(t)=ax(t)+b\int_0^tx(\tau)\text{d}\tau,$$
where $\dot{x}(t)$ denotes the time derivative of $x(t)$. If we derive the above equation and ...
1
vote
1answer
28 views
Differential equations, convergence?
I am dealing with the following matrix.
$A=\begin{pmatrix}
0&a & a & a & c & c &c & c\\
a& 0& a &a & c& c& c& c\\
a& a &0 &...
0
votes
1answer
39 views
Reason for choice of the word “asymptotically” stable in Lyapunov stability theory?
The equilibrium $x^\ast$ is (Lyapunov) stable iff $$\forall \varepsilon > 0 \; \exists \delta(\varepsilon) : \lvert x(0) - x^\ast \rvert < \delta(\varepsilon) \Rightarrow \forall t \geqslant 0 \;...
1
vote
0answers
33 views
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 ...
2
votes
1answer
92 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 = ...
0
votes
0answers
23 views
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 ...
0
votes
0answers
16 views
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 ...
0
votes
0answers
49 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? ...
3
votes
1answer
53 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 ...
1
vote
1answer
17 views
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 ...
1
vote
0answers
48 views
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$, ...
2
votes
0answers
20 views
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(...
2
votes
0answers
49 views
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 ...
2
votes
0answers
41 views
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 ...
0
votes
0answers
52 views
Stability of ODE involving trig functions and nonhyperbolic fixed points
Consider the following autonomous vector field:
$$\dot x = −x$$
$$\dot y = \sin y$$
where $x \in \mathbb{R}^2, -\pi ≤ y ≤ \pi$
$\bullet$ Find all fixed points.
$\bullet$ Determine the linearized ...
0
votes
1answer
55 views
Stability of a 3D nonlinear ODE/dynamical system
I have tackled many 2D systems, but not any with 3D. I'm convinced that the principles and concepts still hold, however the problem is the lengthy computation of the eigenvalues as we have a 3x3 ...
2
votes
0answers
61 views
Hartman-Grobman theorem on a manifold
Consider a dynamical system $\Sigma$ on a manifold $M$ of dimension $d$ embedded in $\mathbb{R}^n$, where $d<n$. Let $x\in M$ be an equilibrium point and suppose we wish to determine the stability ...
0
votes
0answers
29 views
How to understand the Floquet Theory?
I am studying a physical case of a circular cylinder vibrating in a quiescent incompressible fluid.
Such a system can be determined by two non-dimensional groups the Keulegan-Carpenter number and ...
1
vote
0answers
80 views
Center manifold of nonhyperbolic fixed point
Question: Consider the following autonomous vector field on the plane:
$$\dot x = −x $$
$$\dot y = −x^2$$
$$(x,y) \in \mathbb{R}^2$$
$\bullet$ Compute the flow generated by this vector field.
$\bullet$...
1
vote
0answers
36 views
How to describe a system with stability in terms of some variables
Assume I have a system:
$\dot{\mathbf{x}}=-\frac{\partial}{\partial\mathbf{x}}L(\mathbf{x},\mathbf{y}) \\ \dot{\mathbf{y}}=\frac{\partial}{\partial\mathbf{y}}L(\mathbf{x},\mathbf{y})$
where $\...
0
votes
0answers
66 views
Conditions for which all orbits are periodic through varying constants.
Question: Consider the following autonomous vector field on the plane:
$$\dot x = ax+by$$
$$\dot y = cx+dy $$
$$(x,y) \in \mathbb{R}^2 \qquad (a, b, c, d) \in \mathbb{R}$$
$\bullet$ Give a set of ...
1
vote
2answers
38 views
How do you obtain the fixed points and stability of a piecewise function?
Hi so I'm trying to work out how to find the fixed points of a piecewise function and do a stability analysis. One of the past exam papers in this complex systems course I'm doing has this question:
...
0
votes
0answers
18 views
If a time-varying discrete system matrix is bounded by two constant Schur stable matrices, what does it imply?
Consider a time-varying discrete system matrix $A_{t}$ that is bounded by
two constant Schur stable matrices as follows
\begin{eqnarray*}
B & \leq A_{t}\leq & C,
\end{eqnarray*}
where the ...
1
vote
0answers
35 views
Why is stability of D.E. defined on infinite interval?
I will cite the following definition (from an O.D.E textbook) of the stability of a solution:
Definition. Let $f:[0,\infty)\times\Omega\to\mathbb{R}^n$ be continuous and locally Lipschitz on $\Omega\...
2
votes
1answer
44 views
Instability of a system subject to periodic perturbation (cont'd)
This is a follow-up to this question.
Consider the following 2-dimensional system
$$
\dot{x}(t) = A(t)x(t) \quad x(0)\in\mathbb{R}^2,
$$
where $A(t)$ is a 2-dimensional time-varying matrix. Suppose ...
0
votes
0answers
79 views
Verifying stability of equilibria directly from the flow of an ODE
Question: Consider the following autonomous vector field on $\mathbb{R}$:
$\dot x = x-x^3, x \in \mathbb{R}$
Compute all equilibria and determine their stability, i.e., are they Lyapunov stable, ...
2
votes
1answer
71 views
Instability of a system subject to periodic perturbation
Consider the following 2-dimensional system
$$
\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 ...
