# 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 ...
• 119
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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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### 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 $$\dot{x} = f(t,x) ,$$ 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 $$x''(t) + x(t) = \epsilon \sin(x(t))$$ 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 ...
• 139
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