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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Verify a set is positive invariant of Kuramoto model
Consider $$\frac{d\theta_i}{dt}=-\sum_{i<j}A_{ij}\sin(\theta_i-\theta_j)$$
where $A_{ij}$ is adjacency matrix of a connected graph, and $\theta_i\in\mathbb{R}^n$, $\forall i\in\{1,2,\cdots,n\}$.
...
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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 ...
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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 ...
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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/...
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On exponential stability of fixed points
I am a bit lost in the concepts of stability theory. Consider the (non-linear) ODE $x' = \varphi(x)$ in some Banach space with a unique stationary point $x_*.$
Then we could say that the fixed point ...
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Trapezoidal Stability of differential equations
I am getting confused when attempting to obtain the condition for stability of the differential equation $$\frac{d}{dt}(y)=\alpha$$. Like, the trapezoidal formula
$$y_{n+1}=y_n+\frac1{2}h\left(f(t_n,...
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Strong convexity implies existence of local optimal
I am reading this thesis (2006) by Daniel Wachsmuth on the optimal control of the unsteady Navier-Stokes equations. In Chapter 5: Stability of Optimal controls of this thesis, he use strong convexity ...
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Determine stable set via simultaneous stability
Consider a system of n nonlinear ODE's $\dot{x} = f(x)$, where $x \in \mathbb{R}^{n}$, with only a few equilibrium points scattered around (The one in my mind right now is the Kuramoto model ) Given ...
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Stability of the equilibria
Could somebody help me to prove that the equilibrium is stable?
\begin{equation}\begin{cases}
u'=u(a_1-b_1u+c_1v+r_1w),\\
v'=v[(1-k)a_2+b_2u-c_2v],\\
w'=kb_3v-(r_2u+q)w.\end{cases}\end{equation}
I ...
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Linear Stability Analysis: Maximum Growth Rate
I want to calculate the maximum growth rate from the following dispersion relation:
$$ \omega(k) = \frac{1}{\bar{k}}\bigg(k^4(a+b)+k^2(a+b) \bigg) $$
where $a,b$ are constant expressions dependent on ...
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Intersection Between Stable and Saddle-Point Solutions -Stability Analysis
I have a free energy function:
$$G(N_b, l_b)= -N_b E_b + \frac{1}{2} N_b \kappa_b (l_b - 1)^2 - F(l_b - 1) +
\frac{A}{2} k_g (u_g - l_b)^2 + (N_t - N_b) \text{Log}\big(\frac{N_t - N_b}{A}\big) + ...
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Positive and negative eigenvalues - Saddle points
This is similar to another question on my page but this one is more conceptual: If you have a saddle point, is this always classified as unstable if you're doing a stability analysis? What do they ...
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Saddle Points in a Linear Stability Analysis
I'm working on a linear stability analysis and have reached a quadratic that led me to two eigenvalues of opposite signs.
$$\chi s^2 + \bigg [\frac{\chi v^\infty}{d_c} - \frac{\chi\beta}{\zeta} - k + \...
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Stability of discrete-time dynamical systems using Lyapunov stability
I am studying the use of LMIs as an analysis tool for discrete-time dynamical systems. Consider the autonomous discrete-time system given by
$$
x_{_{k+1}} = A x_{_k} \tag{1} \label{sys}
$$
where $ x \...
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Hurwitz stability status of two matrices
I have a complex symmetric matrix (it is not Hermitian), i.e. $\textbf{A}\in\mathbb{C}^{n\times n}$. Can you prove that $\textbf{A}$ and $\textbf{B}=\textbf{A}+\textbf{A}^*$ have similar Hurwitz ...
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Stability of a real matrix and sign changes
Question: Given a stable $n\times n$ real matrix (namely, all the eigenvalues have strictly negative real part), am I guaranteed that by changing the sign of $i<n$ rows, the matrix will become ...
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Asymptotic properties of a nonlinear difference equation
I have been stuck with this difference equation for a while and have had no success getting a proper discussion on the asymptotic behavior of it or its stability conditions.
Suppose that $(Y_n)_{n\ge ...
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Show any root of an elementary symmetric polynomial does not have positive imaginary part in all it's components
We define $H=\{u \in \mathbb{C}: \text{im}(u)>0\}$ to be the open upper half plane. For $n \in \mathbb{N}$ and $k = 1, \ldots n$ let $e_{n,k}(x_1,\ldots,x_n)=\sum\limits_{1 \leq j_1 < \ldots <...
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Non-escaping property of stable manifold for flows
Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a smooth map and $\bar{x}$ be a fixed point of $F$. Assume that the Jacobian of $F$ at $\bar{x}$ has only eigenvalues with magnitude strictly smaller or larger ...
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How large the error ball of $\|x\| $is, when using the Lyapunov function $x^\top Px$?
I would like to know how large the error ball of $\|x\|$ is when using the Lyapunov function $x^\top Px$:
Assumption: I have an almost linear closed-loop system $\dot{x}=(A-BK)x+\epsilon(x)$ with ...
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Equilibrium points of hybrid function
I'm trying to understand the equilibrium for the below problem. This is what I have - was hoping someone can check for any glaring errors?
Consider the piecewise function $f$ defined by
$\begin{align*}...
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Proof that the push-forward of a sheaf is unstable
Let $\pi:X\to C$ be a (proper) elliptic surface with $F$ as a general fiber and H a polarization on $X$.
Let $\mathcal F$ be a bundle on $C$ and let $\mathcal E:=\pi^* \mathcal F$.
Assume $\mathcal E$ ...
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Find a Lyapunov function and prove that an almost linear closed-loop system is stable
I would like to find a Lyapunov function and prove the following closed-loop system is stable:
$\dot{x}=(A-BK)x+(z(u)+\epsilon)$,
where a function of control input $z(u)$ satisfies $\|z\| < \rho \|...
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How to deal with the Floquet multipliers for a second-order ODEs system?
I am interested in the study of the stability of a system via the Floquet multipliers. So, I understand that when I have a first-order ODEs system each of the variables is of interest. For instance, ...
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Globally exponentially stable point
consider this linear, non-autonomic system:
$x_1 ̇=-x_1-f(t)(x_2-x_3 )$,
$\ x_2 ̇= -x_2+x_1$,
$x_3 ̇=-x_3-x_1$
where $f(t)$ is continuously differentiable and satisfies
$0≤f'(t)≤f(t)≤k$ for all $0≤t ...
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Why solving Lyapunov’s equation proves the stability?
The equation:
A'P + PA = -Q
where A is an n×n matrix, P is a symmetric positive-definite matrix of the same size, and Q is a symmetric positive-semidefinite matrix.
A is the matrix of the system.
I am ...
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Time Derivative of Dynamical System
Suppose I have a dynamical system of the form
$$
\frac{dx}{dt} = f(x)
$$
Most of the frameworks I am familiar with for analyzing such systems revolve around finding the fixed points $x^*$ where $f(x^*)...
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Hint for showing that the equilibrium of a nonlinear system is a center
Consider the second-order nonlinear dynamical system
\begin{align*}
\dot{x}&=x^3-y\\
\dot{y}&=x-x^2y
\end{align*}
The (0,0) equilibrium is obviously a center but I cannot find a way to prove ...
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Show that $\dot{x}(t)=a(t)x$ Uniformly Stable when $a(t)\in C(t_0,\infty)$
Consider the scalar equation $$\dot{x}(t)=a(t)x(t)$$ which is stable and reducible.
I'm trying to show that $\dot{x}(t)=a(t)x(t)$ is uniformly stable when $a(t)\in C(t_0,\infty)$.
The precise ...
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What is this inequality referred to as? (related to different solutions of the system)
What is this inequality referred to as?
$$\overline{\lim\limits_{t\rightarrow\infty}}\varphi\left ( t \right )- \overline{\lim\limits_{t\rightarrow\infty}}\psi\left ( t \right )\leq\overline{\lim\...
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Understanding proof of stability in an stochastic differential equation
I am currently doing my thesis on stochastic models applied to interest rates. I am partly basing myself on the article "Stability Behavior of Some Well-Known
Stochastic Financial Models" of ...
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How to find the steady state in this system when one variable is irrelevant for equilibrium?
I have a nonlinear system of differential equations for functions $x(t)$ and $y(t)$ with parameters $\epsilon$ and $\lambda$
$$\frac{\text dx}{\text dt}=(1-x)\cdot[1-\epsilon x-2\lambda xy(1-x)]$$
$$\...
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Finding Equilibrium points and testing stability in a 2d system of differential equations
I am studying the a dynamic system wherein the velocity of individuals in an area are represented by ordinary autonomous differential equations that generally looks like this:
$$
m_i\left(\frac{dv_i(t)...
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Is linearisation of ODE around a stable equilibrium always justified?
Let $f:\mathbb R^n\to \mathbb R^n$ smooth. Let $\hat y\in\mathbb R^n$ be a stable equilibrium point for the ODE $y'(t)=f(y(t))$. Namely :
$f(\hat y)=0$,
for every $\epsilon>0$ there exists $\...
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Crossing the limit cycle in a DDE
I have the following delay differential equation
$$
\begin{align}
\frac{dx_1(t)}{dt} &= \frac{1}{1 + \left(\frac{x_2(t-\tau)}{p_0}\right)^n} - \mu_m \cdot x_1(t)\\
\frac{dx_2(t)}{dt} &=...
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How can I relax and stabilise these difficult boundary conditions so I could numerically solve my system of PDEs?
I have the following set of non-dimensional equations that I am trying to solve using the finite difference method:
$$ u_{rr} + {1 \over r} u_r+u_{zz}-{1\over c^2}u_{tt}-f=0 \tag 1$$
$$ f_{rr} + {1 \...
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Asymptotical stability vs Asymptotic uniform stability for nonautonomous systems
Slotine:
For autonomous systems:
For nonautonomous systems:
The second definition says there is a ball $0<R_2<R_1$, where trajectories that start in $R_1$ will converge into the smaller ball $...
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Does reversing the polynomial preserve the number of the roots in the right-hand-side of the complex plane? If so, why?
The Routh-Hurwitz Stability Criterion is essentially an algorithm to determine how many roots a polynomial has in the right-hand-side of the complex plane (that is, how many of its roots have positive ...
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Classification of fixed points in 4D system of autonomous ODEs
Let's say I have a 4D system of autonomous ODEs
\begin{equation}
\begin{split}
\dot{u} = f(u,v,w,z)\\
\dot{v} = g(u,v,w,z)\\
\dot{w} = h(u,v,w,z)\\
\dot{z} = i(u,v,w,z)\\
\end{split}
\end{equation}
...
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Identify the Bifurcation of a map
This question is taken from Glendinning's textbook "Stability, instability and chaos": Given the map $x_{n+1} = \mu - x_n^2$, determine the type of bifurcation which occurs at $\mu = -\frac{...
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Global continuability and smooth deformations
I am currently studying a paper by J.A. Yorke and K.T. Alligood, "Families of Periodic Orbits: Virtual Periods and Global Continuability", about how families of certain periodic trajectories ...
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Procedure for randomly creating linear dynamical systems with stable dynamics [closed]
I want a procedure for randomly generating a square matrix A such that the linear system
x_t+1 = A x_t
is globally ...
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Using linearisation to determine stability of equilibria.
I have been given the system:
$$\begin{cases}\dot{x}=-x+y^2\\\\
\dot{y}=x^2-y\end{cases}$$
(Note: the left hand sides should be x and y with a dot on top however I can't quite find how to write that, ...
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Prove unstable singularity is limit of some solution when $t\to-\infty$.
Consider an ODE $x'=f(x)$, where $f:U\to\mathbb{R}^n$ is $C^1$ and $U$ is some open subset of $R^n$.
Supose $x^*\in U$ is an unstable singularity of the ODE, that is:
$f(x^*)=0$ and there is a ...
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How can I use the Von Neumann stability analysis on a system of finite difference equations?
I am trying to numerically solve the following system of partial differential equations:
$$ k^2u_{xx} - u_{tt} = f \tag 1$$
$$ c^2f_{xx} - f_{tt} = 0 \tag 2$$
I am using the method of finite ...
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Proving there is an assymptotical solution when Jacobian at singularity has a negative eigenvalue
Consider the ODE $x'(t)=f(x(t))$ (in an open subset of $\mathbb{R}^n$) with a singularity at $x_0\in\mathbb{R}^n$.
I want to prove that if the jacobian matrix $A=f'(x_0)$ has an eigenvalue with ...
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Examples of possible stability classification in the critical case
Given an ODE $x'=f(x)$ (in a open subset of $\mathbb{R}^n$) and a singularity $x_0$ (point where $f(x_0)=0$), I learned the following sufficient conditions for classifying stability:
Let $A=f'(x_0)$ ...
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Struggle understanding stability
I am a bit confused about the connection between linear stability analysis, bifurcation points and amplitude expansions. I have a non-linear system, given as, for $1\leq i \leq n$,
$$
\frac{d}{dt}\...
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Normal Mode Analysis - Which theorem states "Bounded domain implies complete orthonormal basis"?
I am reading about normal mode analysis - the idea that perturbations from a steady state can be expanded as a Fourier Series if the domain is bounded, or as a Fourier transform integral in the case ...
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Bifurication curve and stability
Could you please explain for me this bifurication curve? Which part are stable and why?
Many thanks in advance enter image description here
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