# 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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### Will a linear transformation of an ODE result preserve the convergence?

I have an ODE in $x$ defined over the strictly positive part of the simplex, meaning: $\sum_{j=1}^n x_j=1$ and that all $x_j$ are strictly positive. I use the following linear transformation in order ...
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### How can I prove the stability of this system? [closed]

Consider the following system: $\dot{x}_1=x_2$ $\dot{x}_2=-x_2-sgn(x_1)$ where sgn(.) is the sign function. In simulations, both $x$ and its derivative converge to zero but I can not prove it ...
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### Problems computing the Lyapunov spectrum depending on time interval T

I intent to determine the Lyapunov spectrum of nonlinear dynamic systems. To do so, I implemented some code in MATLAB, based on algorithms from literature and very similar to the way, described in ...
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### Proof of Stability of parabolic-like Convolutional Residual Neural Networks

I am trying to understand a recent publication by Lars Ruthotto and Eldad Haber called Deep Neural Networks motivated by Partial Differential Equations published in the Journal of Mathematical Imaging ...
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### Problem understanding bifurcations

I am beginning to study bifurcations, and I have some preoblems understanding some concepts. I have understood that a Bifurcation can be defined has the change of behaviour of a dynamical system as a ...
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### Determine stability of the equilibrium state

Please help me. I am struggling to determine the stability of the equilibrium state $x = 0$ of the system $$x_1' = x_1(x_1^2 + x_2^2 - \beta^2) + x_2 \\ x_2' = x_2(x_1^2 + x_2^2 - \beta^2) - x_1$$ ...
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