# 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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### Is it possible that a non-autonomous non-linear differential equation can be transferred into an autonomous? [closed]

Is it possible that a non-autonomous non-linear differential equation can be transferred into an autonomous? a) Give an example. b) what implications would it be to use one or the other scheme.
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### Logistic model feasability implies stability [closed]

I have a model with logistic grown, as the lotka-volterra with limiting resources by carrying capacity. Does any one knows a theorem which says that in the logistic model, the feasability of the non ...
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### Linear stability analysis of the nonlinear Schrödinger's equation

I have the following equation - $u_z = \frac{i}{2}u_{tt} + i|u|^2u$ where the subscripts denote partial derivatives with respect to the corresponding variable. I will be doing the stability analysis ...
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### How to find a Lyapunov function in this case?

We have the system of differential equations \begin{aligned} \frac{dx}{dt} &= y + \sin{x}\\ \frac{dy}{dt} &= -5x-2y. \end{aligned} It's necessary to prove that the system is stable using ...
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### Finding a Lyapunov function and proving the stability

I'm trying to find a Lyapunov function for $(0, 0)$ in the system \begin{cases} x' = 2x - 2y - (2x - y)^3\\ y' = 4x - 2y + (x - y)^3 \end{cases} I thought that the following one would ...
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### Question about stability of non isolated equilibrium

Suppose I am studying the stability of a nonlinear system using the Lyapunov inverse theorem, and suppose I get a jacobian matrix that is singular. From the theory, I know that this implies that the ...
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### Stability of stationary points of an ODE system

I have the next bidimensional system $$x'=x-2y+x^2-y^2$$ $$y'=-2x+y+3x^2-3y^2$$ The stationary points are $(0,0)$ and $(\frac{7}{8},\frac{5}{8})$, I tried construct a Lyapunov function but I'm stuck. ...
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### How to analyze the derivative of a Lyapunov function

I am studying Lyapunov stability. My question is really short, and it is: Suppose I get a derivative of a Lyapunov function of the form: $\dot{V}(x,y)=-x^5-y^2$ what can I conclude about the stability ...
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### What does $\operatorname{Tr}>0$ imply about complex eigenvalues of a Jacobi matrix?

I am currently taking a mathematical biology course and am working through the notes. We are covering stability analysis using the Jacobian matrix. One of the conditions for checking the stability is ...
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### Are there Lyapunov stability conditions that focus on a subset of the available state equations?

Are there Lyapunov stability conditions that focus on a subset of the available state equations? Essentially, I have a system of nonlinear ODEs and I only care that some of the variables converge to ...
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### Asymptotically stability of positive system

I currently study control theory and read about positive linear systems. In Lorenzo Farina and Sergio Rinaldi's book: "Positive Linear Systems: Theory and Applications", chapter 5 about ...
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### Is every fixed point in an attractor stable?

Let $\Omega$ be a topological space and $\tau:\Omega\to\Omega$. We say that a fixed point $x_0\in\Omega$ of $\tau$ is stable if for every neighborhood $V$ of $x$, there is a neigheiborhood $U$ of $x$ ...
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### Is a fixed point $x^\ast$ of a map $\tau$ with $|\tau'(x^\ast)|>1$ asymptotically unstable?

Let $\tau\in C^1(\mathbb R)$, $x_0\in\mathbb R$ and $$x_n:=\tau(x_{n-1})\;\;\;\text{for }n\in\mathbb N.$$ I was able to show that if $x^\ast$ is a fixed point of $\tau$ with $|\tau'(x^\ast)|<1$, ...
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### Stable and unstable manifold of a system driven by the logistic map

Let $\lambda\in[0,4]$ and $$\tau(x):=\lambda x(1-x)\;\;\;\text{for }x\in[0,4].$$ Noting that $$\tau'(x)=\lambda(1-2x)=0\Leftrightarrow\lambda=0\vee x=\frac12\tag1$$ and $$\tau'(x)=-2\lambda\tag2,$$ it'...