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Questions tagged [stability-in-odes]

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

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Autonomous ODE $\dot{x}=f(x)$: $\lim_{t\rightarrow\infty}x(t)=x^*\Rightarrow f(x^*)=0$

Let $x : [0,\infty) \to \mathbb{R}^d$ be a solution for the autonomous ODE $$\dot{x} = f(x)$$ where $f : \mathbb{R}^d \to \mathbb{R}^d$ is a Lipschitz continuous vector field. We know that $$\lim\...
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Counterexample of Lyapunov' Stability Theorem for equilbrium

I know that thanks to Lyapunov' Stability Theorem one can study the stability of an equilibrium by finding a Lyapunov function associated to that equilibrium. But, is the converse true? Is it true ...
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How can this system not be asymptotically stable?

I am currently studying stability of nonautonomous systems using the book Applied Nonlinear Control by Slotine & Li. On page 125, there is example 4.13: $$ \begin{align} \dot{e} &= -e + \...
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18 views

Stability and attraction properties of ODE does not depend on initial condition

According to my lecture notes: The properties of stability or attraction do not depend on the chosen initial time $t_0$. This is a consequence of the theorem of continuous dependence with respect ...
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39 views

Unstable numerical solution of ODEs and PDEs

Choosing the right step size for a stiff ODE or a non-linear ODE and PDEs is an important factor. While studying a paper on choosing appropriate step size in numerically solving ODE, I questioned: ...
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Trying to understand the definition of Lyapunov stability for periodic solutions

Consider the following equation $$ \dot{x} = f(t,x)\tag{5.1} $$ and the following definition Definition (Stability in the sense of Lyapunov for periodic solutions): Consider equation $(5.1)$ with ...
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51 views

How do Static Loop-Transformations Work

Loop-Transformations (input feedforward or output feedback) can be used for passivity analysis, also in the case when considering a static (memoryless) nonlinearity. This is described for example in ...
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2answers
33 views

Showing that equilibrium is unstable

I just solved an ODE $x'=A(t)x$ which has $x(t)=e^{t/2}(-\cos t, \sin t)^T$ as a solution. Now I want to show that the equilibrium $\bar x=0$ is unstable. So according to my book I need to show that ...
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42 views

Definition of stable manifold

The text I am using "Nonlinear PDEs - A Dynamical Systems Approach" (Hannes Uecker) defines a stable manifold as follows Definition : Let $u^*$ be a fixed point of the ODE $\dot{u} = f(u)$ with ...
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Showing a scheme is convergent through Von Neumann stability analysis

Show that the box scheme $$\frac{1}{2k}\Big[(U_j^{n+1}+U_{j+1}^{n+1})-(U_j^n + U_{j+1}^n)\Big]+\frac{a}{2h}\Big[(U_{j+1}^{n+1}-U_{j}^{n+1})+(U_{j+1}^n - U_{j}^n)\Big]$$ is convergent for the ...
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Stability of the system without equilibrium points [closed]

What can be said about the stability of the system if it does not have an equilibrium point ? e.g. $x' = x^2 + y^2 + 1, \tag{1}$ $y' = x^2 - y^2, \tag{2}$
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1answer
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Stability of equilibrium of a nonlinear system of ODE's

Suppose we have the nonlinear system of ODE's $$\begin{cases} \dot{x_1} = -\beta x_1 x_2 \\ \dot{x_2} = \beta x_1 x_2 - \gamma x_2 \end{cases} $$ Where we take $\beta, \gamma > 0$ arbitrary for ...
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Stability and asymptotic stability of the equilibrium point $\tilde{y}$=0

Consider the ODE:$$ y'= sin(y)$$I have to investigate the stability and asymptotic stability of the equilibrium point $\tilde{y}$=0. Now we have defined an equilibrium point stable, in the sense of ...
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70 views

Finding the local center manifold

Question: Consider the system \begin{align} \frac{dx}{dt} & = x^2+xy+y^2 \\ \frac{dy}{dt} & = x^2+xy-y \end{align} Find a quadratic approximation for the center manifold about $(0,0)$. Find ...
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Sketching phase diagrams and typical solutions with small perturbations

a.) Sketch a phase diagram for $x' = x^3-2x^2+x$. Label all equilibrium solutions as stable, unstable, or semi-stable b.) How does the phase diagram change if we consider instead $x' = x^3 - 2x^2 ...
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Bifurcation plot - explanation desired

I have a two variable circuit: $$\frac{dx}{dt} = -x + \frac{1}{1+Ky}, \hspace{2mm} \frac{dy}{dt} = 1 - (a+bx^2) y$$ I have already shown that conditions of $a$ and $b$ for a bifurcation are $$\frac{a}...
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Proving the existence and stability of a periodic solution

I want to prove the existence of a periodic solution for this differential $$ \frac{dN}{dt}=(1+cos(\alpha t))r(c-N)-\mu N. $$ Plotting the solution of this equation in Matlab produces the graph ...
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51 views

Energy-Casimir method

In the Energy-Casimir method, it is said we look for 'conserved quantities' that are a function of the system. Then subsequently $V = \mathcal{H} + \mathcal{C}$ becomes the candidate Lyapunov function,...
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28 views

conditions for asymptotic comparison to hold

I have the following simple dynamical system: \begin{align} x_1' &= a - f(x_2)x_1\\ x_2' &= bx_1 - cx_2, \end{align} where all parameters and initial conditions are positive. $f(x_2)$ is a ...
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What are some methods for finding global stability for this 3D system?

I am studying some biology system and arrives at this simplified dynamics: \begin{align} x_1' &= a_1 + a_2x_2 - a_1x_2 - a_4x_1 - a_5\frac{1+a_6x_3}{1+a_7x_3}x_1\\ x_2' &= a_5\frac{1+a_6x_3}{...
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2answers
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Using a Lyapunov function to classify stability and sketching a phase portrait

Consider the system $$x' = -x^3-xy^{2k}$$ $$y' = -y^3-x^{2k}y$$ Where $k$ is a given positive integer. a.) Find and classify according to stability the equilibrium solutions. $\it{Hint:...
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Global stability question for system with a unique locally-asymptotically-stable steady state

I have an ordinary differential system of dimension larger than 2 that contains a locally-asymptotically-stable unique fixed point. Additionally, the system is bounded. Now, suppose that I can show ...
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1answer
69 views

Using a Lyapunov function to determine stability of equilibria

Given $$\left\{\begin{aligned} x' &= -x^3 + 7xy^2\\ y' &= -3x^2y+y^3\end{aligned}\right.$$ find $a, b > 0$ such that $L(x,y) = a x^2 + b y^2$ obeys $\frac{d}{dt}L \neq 0$ whenever ...
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17 views

Solving the normal mode equations of a 2D linear elastic solid

I am trying to solve the linear stability of a 2D elastic solid whose dynamics are given by the Navier equations, $\frac{\partial}{\partial x} \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\...
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72 views

Lyaponov function in dynamic system (in polar cordonates)

Let the following system in $R^2$ \begin{equation} (S) \left\{ \begin{array}{l c } \overset{.}{\rho}=\rho(1-\rho) \\ \overset{.}{\theta}=\sin^2(\frac{\theta}{2}) \end{array} \right. \end{equation} ...
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1answer
104 views

Requirement of Lyapunov Stability in Asymptotic Stability

In my Differential Equations course, we defined the equilibrium point $x_0$ of a dynamical system $\frac{dx}{dt} = f(x(t))$ (for $f$ defined on an open subset of $\mathbb R^n$, say $\mathbb R^n$ ...
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2answers
50 views

Having trouble solving this separable differential equation

I am having some trouble with the following separable differential equation $$\frac{dx}{dt} = x(x-1)(x-3)$$ with initial condition $x(0) = 2$. What is $\displaystyle\lim_{t \to \infty} x(t)$? I am ...
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1answer
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Stability matrix - order of elements

I have two equations: $$\frac{dx}{dt} = v(k-ux) - \delta x, \hspace{3mm} \frac{dv}{dt} = v(r-px)$$ I want to calculate the Jacobean so I can analyse the stability of the fixed points. Does it ...
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1answer
28 views

Non-linear system of ODE:s - Maximal existance interval.

A modell for two competing species is the non-linear system of differentialequations \begin{align} y'_1(t) &= s_1 y_1(t)\left(1-\frac{y_1(t)}{N_1}\right)-a_1y_1(t)y_2(t)\\ y'_2(t) &= s_2 y_2(...
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What are hydrodynamic and non-hydrodynamic modes in the Navier-Stokes equations and how do they relate to stability?

I'm new here. I have been reading articles on Navier-Stokes equation and the stability analysis by studying different modes. However I am still confused by what is meant by modes as in the context of ...
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Prove existence of a Heteroclinic Orbit

How would you go around proving that there exists a heteroclinic orbit between two equilibria ( in the problem I'm trying to solve, one is a stable node(say n) and the other is a saddle (say s))? I ...
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Showing the equilibrium point to be globally exponentially stable using Lyapunov indirect method.

We have the system $\ddot{q} + \dot{q} + g(\dot{q},q) + q = 0, \forall t \geq 0$. $x = \begin{bmatrix} x_{1}\\ x_{2}\end{bmatrix} = \begin{bmatrix} q\\ \dot{q}\end{bmatrix}$ $\dot{x} = Ax + h(x)$ ...
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How to understand assumption from stability theorem?

Consider $$x'(t)=A\cdot x(t)+f(t,x(t)),\,\,x(0)=a$$ for $t\geq0$. Assume that $A$ is real $N\times N$ matrix with negative real parts of eigenvalues, $f\colon \mathbb{R}\times \mathbb{R}^N\to\mathbb{R}...
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1answer
36 views

Singular perturbation theory in non-standard form

Singular perturbation theory in ODE's is a well treated and highly studied subject. Most of the references I can find take the form, \begin{align} \dot{x} &=f\left( x,z,\varepsilon \right) \\...
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43 views

Critical points of the following ODE

Consider the following nonautonomous, nonlinear ODE: $$y'(t)=\rho(W(t)y(t)+b(t)), \hspace{6pt}t\geq 0,$$ where $y(t),b(t)\in\mathbb{R}^{n},W(t)\in\mathbb{R}^{n,n}$ and $\rho:\mathbb{R}\to\mathbb{R}$ ...
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1answer
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How to adjust the diagonal so that a matrix is on the stability threshold?

I am working on the stability of food webs, which can be represented by a Jacobian matrix showing the interaction strengths between species. I know that a matrix is locally stable if all real parts of ...
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0answers
50 views

Stability of the following ODE

Consider the following nonautonomous, nonlinear ODE: $$y'(t)=\rho(W(t)y(t)+b(t)),$$ where $y(t),b(t)\in\mathbb{R}^{n},W(t)\in\mathbb{R}^{n,n}$ and $\rho:\mathbb{R}\to\mathbb{R}$ is some nonlinear ...
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59 views

How fast does a solution approach an equilibrium?

I have an autonomous dynamical system $\dot{\mathbf x}(t)=\mathbf f(\mathbf x(t))$ on $\mathbb R^2$, and I found that solutions are future asymptotic to an equilibrium point $\mathbf a$, i.e. $\lim_{...
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30 views

Globally uniformly asymptotic stability proof

In dynamics theory, here is a question A large class of time-varying capacitor-linear resistor networks can be described by equations of the form $$x'_i=-\sum_{j=1}^n[a_{ij}\,d_{1j}(t)+b_{ij}\,d_{...
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What is a slow manifold, and why is it called slow?

I am studying a dynamical system which has an equilibrium point where the linearisation of the system at that point exhibits one eigenvalue which is exactly 0. According to Wikipedia that means that ...
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1answer
18 views

Finite Element in space, finite difference in time, stability analysis

I was wondering how to do stability analysis for the classical wave equation if we discretize in space with finite element and step forward in time using some explicit scheme like for example leapfrog ...
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38 views

Dynamical system - System of non-linear 1st order autonomous ODEs: future stability

I am looking at a dynamical system of the following form (prime denotes derivation with respect to t): $H' = -(1+2S^2+A(3\sin(\alpha)^2-1))H$ $S' = -\big(2(1-S^2)-A(3\sin(\alpha)^2-1)\big)S+1-S^2-A$ ...
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1answer
79 views

Proving that a system of ODEs has a focus at the origin

Prove that the following system of ODEs has a focus at the origin. $$\begin{aligned} \dot{x} &= -x^3-y^3\\ \dot{y} &= x^3 \end{aligned}$$ Plotting the vector ...
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50 views

Direction Field in MatLab

I need to plot the direction field and phase plane of the following ODE using matlab. I've tried using meshgrid w/ the quiver function, however, I'm not getting the correct field. I need to plot $\...
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1answer
39 views

Stability of the RC Circuit with Two Capacitors

I want to find poles and zeros of the circuit shown below. So I can determine stability of the system. $$ V_{C1} + V_{C2} + V_R = 0 $$ where $I$ is current of the loop, $$ V_{C1} + V_{C2} + RI = 0 $$...
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Dynamical systems with forced oscillation

I want to work out the dynamics of this equation (by that I mean find the fixed points of the system and then the stability of the fixed points): $\frac{dN}{dt}= \sin(\alpha t)\left(rN\left(c-N\right)...
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44 views

Different kinds of stability that apply to planar periodic orbits, and what do they mean?

This is a question about terminology related to orbit stability. I had wanted to ask about stability of orbits described in the paper Three Classes of Newtonian Three-Body Planar Periodic Orbits ...
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77 views

stability analysis of an ODE

I need help on how to linearize the following ODE equation so that I am able to do Stability analysis for the equation. Thanks for the help. $\frac{dQ}{dz} = 2aM^{1/2}$ $\frac{dM}{dz} = \frac{QF}{...
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1answer
24 views

Finding the absolute stability

I am trying to find whether the following is stable absolutely using the improved Euler and the Adams-Bashforth 2 scheme, $u'=\begin{bmatrix} -20&0&0\\ 20&-1&0\\0&1&0\end{...
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23 views

Can one use a Lyapunov-function to prove that an equilibrium is unstable?

Is there a way to construct an argument that uses a Lyapunov-function to show that an equilibrium is unstable? Let's consider the following example: $$\begin{align}\dot x&=y+\epsilon(x^3+2xy^2)\\...