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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Asymptotic stability and Lyapunov functions
I fail to understand a passage in the proof of the following theorem (right after the definition that gives the context of my question):
(Definition of Lyapunov function)
Let $\Omega$ be a ...
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Given is the system of differential equations
Given is the system of differential equations:
$$\begin{cases}\dot x=4y \\ \dot y=-3x \end{cases}$$
(a) Write the first integral of the system. Is the system conservative? Explain.
(b) Sketch the ...
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Mathematic Modeling [closed]
How to solve the following equation manually?
I want to find the equilibrium point of the system of equations below
https://drive.google.com/file/d/1gPUU3N2m1b4tuhxUHaRFHRaOv8gll7_c/view?usp=sharing
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How to simplify the following stability criteria?
I'm trying to understand the mathematics explained in the following video. Basically, we want to identify the constraints on the parameters $k_1$ $k_2$ and $T$ to have a stable second-order system ...
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Lyapunov stability of a periodic system
Consider a planar system
$$
\begin{cases}
\dfrac{\mathrm{d}x}{\mathrm{d}t}=-y,\\
\dfrac{\mathrm{d}y}{\mathrm{d}t}=(a+\varepsilon\cos t)x,
\end{cases}
$$
where $a>0$ and $a\notin\{\dfrac{n^2}{4}\ |\ ...
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Asymptotical stability and stability for homogenous ODE
Let $A \in \mathbb{R}^n$. Prove, that the zero solution of $x'=Ax$ is
Asymptotically stable $\iff$ $Re(\lambda)<0$ for all eigenvalues $\lambda$ of matrix A
Stable $\iff$ $Re(\lambda) \leq 0$ and ...
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Stability of Hamiltonian system on degenerate critical point.
I'm trying to find information on the stability of the following ODE:
$$ x'' = x^4-x^2.$$
We know that it has a Hamiltonian $H(x,y) = \dfrac{y^2}{2} - (\dfrac{x^5}{5} - \dfrac{x^3}{3})$. The orbits ...
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Connection between Liouville's formula and stability of linear systems
In my university textbook I have a statement:
For a linear differential system $\frac{dx}{dt}=A(t)*x$ to be stable is necessary that $\int_{0}^{t} SpA(\tau)d\tau <= M$,
where $M$ is a constant, $...
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Stability of stationary points
In the article by Hirsch "On stability of stationary points of transformation groups
It's mentioned that $0$ is a stable stationary point of the diffeomorphism $f(x)=x+x^3$ (stationary point of ...
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Question about the proof that uniform asymptotic stability can be characterized by KL function. (Lemma 4.5 in Nonlinear Systems (3rd) by Khalil)
Lemma 4.5 in Nonlinear Systems (3rd):
Consider the nonautonomous system
\begin{equation}
\dot{x} = f(t,x) ,
\end{equation} where $f : [0,\infty) \times D \to \mathbb{R}^n$ is piecewise ...
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Exercise about stability and inequality in ODE.
I am stuck with this exercise.
Let
\begin{equation}
x''(t) + x(t) = \epsilon \sin(x(t))
\end{equation}
with initial conditions $ x(0) = x_0, \ x'(0) = v_0 $.
1. Write the problem as a first ...
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Confusion about stability of PDEs.
I have been reading about Stability Theory and have been left with some questions at is seems to me that some of its notions are not very well-defined or at least inconsistently used.
Consider the ...
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Lyapunov stability in a one-sided neighborhood?
Consider a switching system
$$ \dot x =
{ - x, \quad {\rm{if}}\quad x \ge 0} $$
$$
\dot x =
{v \left( t \right), {\rm{ if}}\quad x < 0}
$$
where $ v(t) $ is bounded but indefinite (can be ...
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Studying stability of pde
I have a problem with studying the stability of this PDE.
$$U_t = U_{xx} + f(U).$$
Let $U^{*}(x)$ be a solution for this equation. As conditions we have ${U^{*}}'(x) > 0$ for $x < x_{0}$, ${U^{*}...
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Stability estimate for elliptic reaction-diffusion problem with inhomogeneous Neumann boundary conditions?
I want a stability estimate for the problem $$-\Delta u + u = f \text{ on } \Omega,$$
$$\nabla u \cdot n = g \text{ on } \partial \Omega,$$ where $g \in L^2(\partial \Omega)$.
The problem has ...
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