# 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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### 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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### 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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### 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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