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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19 views
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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votes
1answer
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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1answer
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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2answers
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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votes
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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1answer
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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0answers
29 views
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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0answers
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
$$
q_{in}(t) - q_{out}(t) = A\dfrac{dh}{dt} \quad \text{,}
$$
...
1
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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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0answers
55 views
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:
$$\dot{x_1}(t)=-x_1(t) + x_2(t)[...
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votes
1answer
59 views
Properties of Positive Real Functions
I am trying to understand the properties of positive real (PR) and strictly positive real (SPR) transfer functions. If given a transfer function I know how to determine whether or not the function is ...
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0answers
32 views
Stability of a critical point for a general system ODE
Given an ODE-system I have proven that the solution $\textbf{x}(t)$ will have an upper bound of $\max\{|c_1|,|c_2|\}(\|\textbf{v}_1 \|+\| \textbf{v}_2\|)$. I then want to show that the solution is ...
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0answers
33 views
Dynamics $\delta x(t)=\delta x(0) e^{\lambda t}$ of Henon Attractor
Recall the question I asked before:
Linearized perturbation dynamics of Henon Attractor
So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$\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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0answers
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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0answers
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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0answers
32 views
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 $stp(a/A)$ ?
2
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0answers
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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0answers
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 ...
1
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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 $
...
0
votes
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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2answers
68 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
21 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 ...
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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 ...
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0answers
21 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
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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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 ...
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0answers
17 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:
$$ \...
3
votes
2answers
92 views
How to pick a Lyapunov function and prove stability?
I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for autonomous systems.
Say we are given the nonlinear system:
$$\dot{x_1}(t)=-x_1(t) + x_1(t)x_2(t)...
0
votes
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 ...
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 $$\ddot{\theta}+sin(\...
1
vote
1answer
81 views
Implication of stability of Van der Pol oscillator.
Consider the homogeneous Van der Pol equation, $\ddot{x} + \mu (x^2-1)\dot{x} + x = 0$, with $\mu>0$. We convert it into a dynamical system, $$\dot{\bf x} = (y, -(x+\mu(x^2-1)y), \ \mathbf{x} \...
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votes
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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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
22 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
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 ...
0
votes
0answers
23 views
How to design a Robust observer for a 2D system
Consider the second order system given by $\dot{x}=Ax+Bw(t)$, where $x\in\mathbb{R}^2$,
$$A = \begin{bmatrix}
{0},{6}\\
{-1} {-6}
\end{bmatrix}, \quad B = \begin{bmatrix} {0}\\{1}\end{bmatrix}...
1
vote
2answers
43 views
Showing that a centre of the 2D linear system $\dot{\mathbf{x}} = A \mathbf x$ is Lyapunov stable
Consider the 2D linear system $\dot{\mathbf{x}} = A \mathbf x$ with $$A = \begin{pmatrix} 0&1\\ -4 & 0\end{pmatrix}.$$ The eigenvalues of this matrix are $\lambda = \pm 2i$, meaning that the ...
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votes
1answer
99 views
What's an example where Lyapunov fails to find the bounds of stability
In linear control theory, a system is stable if and only if if satisfies the Routh–Hurwitz stability criterion, so we can use this to solve for the limits of stability. E.g. you can find the maximum ...
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0answers
29 views
Does differentiating an integro-differential equation results in equivalent stability of the solution?
Consider the following integro-differential equation:
$$\dot{x}(t)=ax(t)+b\int_0^tx(\tau)\text{d}\tau,$$
where $\dot{x}(t)$ denotes the time derivative of $x(t)$. If we derive the above equation and ...
1
vote
1answer
28 views
Differential equations, convergence?
I am dealing with the following matrix.
$A=\begin{pmatrix}
0&a & a & a & c & c &c & c\\
a& 0& a &a & c& c& c& c\\
a& a &0 &...
0
votes
1answer
43 views
Reason for choice of the word “asymptotically” stable in Lyapunov stability theory?
The equilibrium $x^\ast$ is (Lyapunov) stable iff $$\forall \varepsilon > 0 \; \exists \delta(\varepsilon) : \lvert x(0) - x^\ast \rvert < \delta(\varepsilon) \Rightarrow \forall t \geqslant 0 \;...
1
vote
0answers
34 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
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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0answers
24 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 ...
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votes
0answers
17 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 ...
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0answers
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? ...
3
votes
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 ...
1
vote
1answer
19 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 ...
