Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [stability-in-odes]

For questions concerning stability of equilibria and of other solutions of ordinary differential equations and their systems.

Stability of a solution of a differential equation means that a small perturbation of initial data will result in only small (or even vanishing) perturbation of solution at later times. See Stability theory on Wikipedia.

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Why don't these ODEs produce the same result?

I am relatively new to differential equations, and the following problem is confusing me. Consider, for example, the ODE $x'+x=0$ such that $x(0)=1$. This has solution $x(t)=e^{-t}$. But consider an $\epsilon>0$ and the ODE. $$\epsilon x'' + x' + x…
user818261
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System with a Lyapunov function over $\mathbb{R}^n$ but not globally asymptotically stable

I'd like to find an example of a system $\dot{\mathbf{x}} = F(\mathbf{x})$, where $\mathbf{x} = \mathbf{0}$ is an equilibrium point, with a corresponding Lyapunov function $V(\mathbf{x})$ that satisfies: (1) $V(\mathbf{x}) > 0 \; \forall \mathbf{x}…
LGenzelis
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Formal proof of Lyapunov stability

I was trying to solve the question of AeT. on the (local) Lyapunov stability of the origin (non-hyperbolic equilibrium) for the dynamical system $$\dot{x}=-4y+x^2,\\\dot{y}=4x+y^2.\tag{1}$$ The streamplot below indicates that this actually is…
RTJ
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Linear Stability Analysis of ODEs/PDEs

I'm looking for a systematic understanding/approach to linear stability analysis of differential equations. I'm interested in an arbitrary (non-linear) PDE, $\mathcal L u=0$ (or system of PDEs although probably best to start with just one, and most…
bjorne
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Lyapunov stability question from Arnold's trivium

V.I. Arnold put the following question in his Mathematical Trivium: Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization? It puzzled me for a while, since my experience doesn't include such a…
H. M. Šiljak
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Stability for Nonlinear System

I am trying to assess the (Liapunov) stability of the equilibrium at $(0,0)$ of the system \begin{align*} x_1' &= -4x_2 + x_1^2 \\ x_2' &= 4x_1 + x_2^2. \end{align*} I plotted the phase portrait in Mathematica, and it looks like a stable (but not…
ec92
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Stability of linear time-varying systems

Consider the following linear time-varying (LTV) system $$\dot{x} = A(t)x$$ If $A(t)$ satisfies $$\mbox{eig} \left( A(t)+A(t)^{T} \right) < 0$$ then is it sufficient to conclude that the time-varying system is stable? I am looking for references…
jjgarrison
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Why is backward Euler more stable?

I'm new to the idea of solving ODEs using the backward Euler. I have a system which I solve using the Backward Euler (actually backward Euler + Newton's method since I can't find a closed form solution). I understand the math involved in solving…
desperateForBibliography
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Stability when eigenvalues are zero : $x' = -x + y + x^2 + ax^3, \space y' = x - y + x^2 + bxy + y^3$

Exercise : Determine the stability of $O(0,0)$ for the system of differential equations: $$x' = -x + y + x^2 + ax^3$$ $$y' = x - y + x^2 + bxy + y^3$$ Discussion : Finding the matrix of the linearized system (the Jacobian), we get : $$J(x,y)…
Rebellos
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On finding the equilibrium solutions to a system of differential equations

I am asked to find all equilibrium solutions to this system of differential equations: $$\begin{cases} x ' = x^2 + y^2 - 1 \\ y'= x^2 - y^2 \end{cases} $$ and to determine if they are stable, asymptotically stable or unstable. I do not know how to…
Monolite
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Choosing a Lyapunov Function for a Nonlinear system (Cubics)

I am working on being able to recognize appropriate Lyapunov functions to show the stability (or instability) of equilibrium points. I have the following system: $\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 &…
rlh282
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How to interpret complex eigenvectors of the Jacobian matrix of a (linear) dynamical system?

Consider a linear ODE system of the following form: $$ \frac {dx} {dt} = Ax $$ In case $A$ has real eigenvectors, I can interpret them as the directions in which the system will move, if the initial value is already a point on the eigenvector. The…
Jannis
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Stability analysis for ODEs with non constant inputs

For a project, I have to deal with systems of ODE's with non constant input such as: $$\begin{cases}\dot x =I(t)x+x^2\\ \dot y=x\end{cases}$$ where I(t) is a random input (for example). In any case, I don't have $I(t)$ as an explicit function. It…
Ambesh
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Finding a Lyapunov function for a given system

I need to find a Lyapunov function for $(0,0)$ in the system: \begin{cases} x' = -2x^4 + y \\ y' = -2x - 2y^6 \end{cases} Graph built using this tool showed that there should be stability but not asymptotic stability. I'd like to prove that fact by…
mik
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Show that the solution of this (nonlinear) ODE cannot remain bounded as $t\to\infty$

Preliminary properties: Let the state vector $x(t)=[x_1(t),\dots,x_n(t)]^T\in\mathbb{R}^n$ be constrained to the dynamical system $$ \dot{x} = Ax + \begin{bmatrix} \phi_1(x_1) \\ \vdots \\ \phi_n(x_1) \\ \end{bmatrix}, \ \ \ \ x(0) = x_0 $$ where…
FeedbackLooper
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