# 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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### Identifying Bifurcation

I am trying to identify bifurcation of my $3D$ system near $a=6.58$. I am getting trajectories as shown in the picture and I am guessing it is Saddle-Node periodic orbit bifurcation since I am getting ...
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### Checking Positivity

enter image description here I am trying to check positivity of an algebraic expressions where all the variables are positive. Using Mathematica 9.1 version I have got the result showed as follows. ...
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### Lyapunov matrix equation theorem

I know the following Lyapunov's theorem: For any symmetric positive definite matrix $Q$, the following are equivalent 1- The matrix equation $A^T X + X A = -Q$ has a unique solution $X$ that is ...
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### from local stability to global stability

Suppose I have the system $x'=F(x)$ with $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$. I denote by $J(x)$ the Jacobian matrix, that is, $J_{ij}(x)=\partial F_i/\partial x_j (x)$. Suppose I know that for ...
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Cetaev Theorem states that: "Considering the ode $X'=F(X)$, with an equilibrium $x_{0}$. If there is a function $V$: $U_{0} \rightarrow$ IR and a region $\omega$ in $U_{0}$ that contains $x_{0}$,...
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### A delayed differential equation system and its characteristic polynomial

I have the following DDE system: \begin{split} \dot{x}_{1} &= -\mu x_{1}(t) + a_{11}f_{11}(\int_{-\infty}^{t} F(t-s) x_{1}(s-\tau_{1})ds) + a_{12} f_{12}(x_{2}(t-\tau_{1}))) \\...
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### Find a stable numerical solution of differential equation $y''=g(x)y$

I'm attempting to numerically solve this differential equation using MATLAB's ode45, presuming a stable solution, i.e., $y$ remains finite as $x\to+\infty$. ($g(x)$ is a pending function determined by ...
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### Numerical stability of spherical Bessel recurrence relation

I'm working on a homemade code that needs an implementation of spherical Bessel function of the first kind myself. The order is restricted to non-negative integers, and only real argument is required. ...
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### How is this stability bound for the unique interpolant possible? $|s^*(x)|^2\le K(x,x)\|f\|_K\text{cond}_2(G_S)$

If one wants to reconstruct a function $f$ which, we assume is an element of a Hilbertspace $(\mathcal{H}(\Omega,K),(\cdot,\cdot)_K)$ of functions $\Omega\to\mathbb{R}$ with a reproducing Kernel (a.k....
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### Estimate solution of nonlinear differential equation by linear differential inequality [closed]

Consider the linear ordinary differential equation: \begin{align*} \dot x(t) & = Ax(t), \\ x(0) & = x_0, \end{align*} $t\geq 0$, $x\in\mathbb{R}^n$, with $A$ is Hurwitz (i.e., all its ...
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### LaSalle's invariance principle for positive semi definite V

With $\dot{\textbf{x}}=\dot{\textbf{x}}(\textbf{x})$, if I have a function which has $V(\textbf{x})>0$ when $\textbf{x}\neq \textbf{0}$ for some domain of x, and if $\dot{V}(\textbf{x})\leq 0$ for ...
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### Bifurcation Analysis of Non-autonomous system

Suppose we have a matrix equation $$\frac{d}{dx}\mathbf{u} = \mathbf{A}(x) \mathbf{u} + \mathbf{b}.$$ If $\mathbf{A}(x)=\mathbf{A}$ were constant, then one can inspect the eigenvalues of the ...
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### With $A=A^T$, $Ax=b$ is solved backward stably for $x$ yielding computed $y$. Show that $y / | y |$ is close to $x / | x |$.

Suppose $A \in \mathbb{R}^{m \times m}$ is a real symmetric matrix with orthonormal eigenvector basis $\{ \vec{q}_1, \ldots, \vec{q}_m \}$. Let $b \in \mathbb{R}^m$ be expressed in this basis as: \...
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### The relationship between real negative eigenvalues and convergence rate for ODE.

Let $\pmb{\delta}=\pmb{\delta}^\triangle$ be an equilibrium point for the following ODE , \begin{align*} \frac{\partial \pmb{\delta}(t)}{\partial t}=\pmb{F}(\pmb{\delta}) \ with \ \ \pmb{F}(\...
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### Meaning of complex eigenvalues for 2D matrix realtive to dynamical systems

I am studying non-linear dynamical systems with the linearization method around an equilibrium point, but I don't get the geometrical meaning of complex eigenvalues. (Let's focus on a 2D case) For ...
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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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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/...