Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [stability-in-odes]

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

1
vote
1answer
7 views

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 ...
3
votes
0answers
43 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 ...
0
votes
0answers
53 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_{...
0
votes
0answers
24 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_{...
1
vote
0answers
25 views

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 ...
0
votes
1answer
16 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 ...
0
votes
0answers
37 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): $H' = -(1+2S^2+A(3\sin(\alpha)^2-1))H$ $S' = -\big(2(1-S^2)-A(3\sin(\alpha)^2-1)\big)S+1-S^2-A$ ...
2
votes
1answer
77 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 ...
0
votes
0answers
18 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 $\...
0
votes
1answer
34 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, $$ V_{C1} + V_{C2} + RI = 0 $$...
2
votes
0answers
69 views

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)...
2
votes
0answers
42 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 ...
0
votes
0answers
76 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}{...
0
votes
1answer
22 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{...
0
votes
0answers
21 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)\\...
0
votes
3answers
57 views

Inequality manipulation for a vector norm

How can I manipulate the following inequality to reach from $$\dot{V}\leq -4x_1^2 +4x_1x_2 -2x_2^2 $$ to $$\dot{V}\leq -(3-\sqrt{5}) \|x\|^2 $$ where $x=[x_1 \;x_2]^T$ is a 2D vector and $\|x\|$ is ...
0
votes
0answers
34 views

Bifurcation of two parameters $\lambda$,$\mu$

I'm already familiar with bifurcations of differential equations with only $1$ parameter $\lambda$, like: $$ x'=-\lambda x - x^4 $$ but what if we're given a differential equation with $2$ parameters:...
1
vote
0answers
18 views

Lyapunov Indirect Method

I have searched a reference (book/paper) where I could find a theorem related to the Lyapunov Indirect Method for any equilibrium point, but I have not found yet. I only found for the zero equilibrium ...
2
votes
0answers
18 views

Why is A-stability and L-stability tested on exponentials?

Is there a reason why stability of numerical methods is tested on exponentials functions? Can such concept be generalized for other functions? I mean how does testing stability on functions $f(x) = e^{...
0
votes
0answers
18 views

Bifurcation points of system of non linear differential equations

I had a ODE of second order, then I built the following system of differential equations: $$x' = y$$ $$y' = -\lambda sin(x) -2sin(x) -sin(x)cos(x)$$ With $\lambda = \frac{w^2}{\Omega^2}$ . I want ...
0
votes
0answers
45 views

On the physical sample of “ discontinuous differential equations ”

The following source contains several physical examples: https://arxiv.org/pdf/0901.3583.pdf. How can I get the differential equation in "Example 2: Brick on a frictional ramp"? How can I grasp this ...
2
votes
2answers
87 views

Bifurcation points of differential equation (example)

Assume the differential equation: $$ x'=\lambda^2-8a\lambda x+2x^2, \quad a\in \mathbb{R}. $$ The critical points are the solutions to the equation: $$ x'=0 \iff 2x^2-8a\lambda x +\lambda^2=0\tag{1} $$...
1
vote
0answers
66 views

Bifurcation in a linear system with 2 equations and 1parameter

I have the following system $$\frac{dx}{dt} = ax+y$$ and $$\frac{dy}{dt} = -x+ay$$ By setting the derivative equals to $0$, we get the equilibrium point $(0,0)$. After drawing the phase portrait for ...
0
votes
0answers
57 views

When does any solution of an i.v.p. converges to some steady state?

Preliminaries Let $A = [0, 1]^N$. Consider a dynamical system $\dot{x} = f(x)$, where $x = x(t) : \mathbb{R} \to \mathbb{R}^N$ and $f : A \to \mathbb{R}^N$, where each $f_i$ is in the class $\...
0
votes
0answers
35 views

Stability of a critical point for a general system ODE

Given an ODE-system I have proven that the solution $\textbf{x}(t)$ will have an upper bound of $\max\{|c_1|,|c_2|\}(\|\textbf{v}_1 \|+\| \textbf{v}_2\|)$. I then want to show that the solution is ...
-1
votes
1answer
87 views

Stable and unstable fixed points of $y' = 500y^2(1 - y)$

$y' = 500y^2(1 - y)$ has fixed points at $y^*=0$ and $y^*=1$ $y(0) =0$ There are 2 definitions; 1 says that: A fixed point $y^*$ is asymptotically stable if $$ \sigma(Df(y^*)) \subset \Bbb C^...
0
votes
0answers
40 views

Fixed Points and their Stability and Finally Bifurcation diagram

Find and Classify all fixed points and sketch vector field. What's the bifurcation? $$ 𝑥̇ = 𝑟𝑥 − \tan(𝑥)$$
1
vote
2answers
34 views

Stability of $y''+4y'+4y=0$

Decide whether the solution is asymptotically stable, stable but not asymptotically stable, or unstable $y''+4y'+4y=0, \phi(t)=e^{-2t}$ After computation of the eigenvalues I found that it has ...
3
votes
1answer
105 views

Stability of Mathieu equation: $x''(t)+\cos t \,x(t)=0$

The equation $$ x''(t)+\cos t \,x(t)=0 \quad (1) $$ can be transformed to the system: $$\vec{x}'= \begin{pmatrix} 0 & 1\\ -\cos t & 0 \end{pmatrix} \vec{x}=A(t) \cdot x(t) $$ with minimum ...
2
votes
0answers
51 views

Show that this system has at least one unbounded solution as $t \to \infty$

Assume the system $$x'(t)=\begin{pmatrix} \frac12-\cos t & 2 \\ 1 & \frac32+\sin t \end{pmatrix}x(t)=A(t)\cdot x(t)$$ with minimum period: $T=2\pi$. Let $\mu_1,\mu_2$ be its characteristic ...
2
votes
1answer
59 views

LaSalle for time varying systems

I am looking for an explanation, why LaSalles theorem is in general not applicable to time varying systems. Can someone provide an example system with $$ \dot{x}(t) = A(t)x(t) \tag{1} $$ I.e., why ...
1
vote
2answers
46 views

Find one domain of attraction for this system

Assume the system: \begin{align} \begin{pmatrix} x \\ y \\ \end{pmatrix}' &= \begin{pmatrix} -(1-y)x \\ -(1-x)y \\ \end{pmatrix}...
0
votes
0answers
28 views

Stability of an Equilibrium Solution for Delayed Logistic Equation

Hello I am trying to study the stability of the dimensionless form of the logistic equation $\frac{dy}{dt} = \alpha y(1 - y(t - 1))$ Note that this is a dimensionless form of the delayed logistic ...
0
votes
2answers
30 views

The reverse implication in the stability theory

Let us consider the system of differential equation $\dot x=Ax+h(t),$ where A is a constant matrix of dimension $n\geq2$ and $h$ is continuous on $[0,\infty).$ All eigenvalues of $A$ have a negative ...
0
votes
0answers
32 views

Properties of a mapping between two differential equations

Let $A\in\mathbb{R}^{n\times n}$ be a nonsingular matrix, $b\in\mathbb{R}^n$, and $f\colon\mathbb{R}\to\mathbb{R}$ be an arbitrary function. Consider the following differential equation $$\tag{$\ast$}\...
2
votes
2answers
138 views

Does $y'=|y|^a$ have any global solutions?

Assume the differential equation $$ y'=|y|^a $$ My intuition tells me that since it involves an absolute value, there might not be any solutions defined everywhere, except for the case $a=0$, where $...
0
votes
2answers
76 views

Solve ODE for real free falling: $y(x)^2\cdot y^{\prime\prime}(x)]=4\cdot 10^{14}$

I am trying to describe the position of a free falling ball by gravity: if $x$ is the time in seconds, $y$ is the position of the falling ball, $y^{\prime\prime}(x)$ is its acceleration then $$ F=G\...
0
votes
0answers
30 views

finding the nullclines of a two connected ODE's

so I've given the following problem: $dn_1/dt=N(t)n_1-n_1$ $dn_2/dt=N(t)n_2-4n_2$ $N(t)=25-6n_1-3n_2$ so I've found the fixed points of the 2 equations and got: [(0,0),(0,7),(4,0)] now I want to ...
3
votes
1answer
87 views

Solving linear system, finding equilibrium and bifurcation points

For homework, I have to solve the following problem: consider the system of ODE's \begin{equation} x'=-x^2+a \\ y'=-y \end{equation} where the parameter $a$ is a real number. I have to characterize ...
3
votes
1answer
108 views

Solving differential equation with linearization and Lyapunov method

For homework, I have to say something about the stability of the zero solution of the differential equation $v''+v+f(v')=0$, where $f$ is a differentiable function satisfying $f(0)=0$ and $f'\geq0$. ...
0
votes
1answer
93 views

Every solution of $x''+x+x^3=0$ is set on $\mathbb{R}$ (proof)

Show that every solution of $x'' + x + x^3 = 0$ is set on $\mathbb R$. Proof: If we transform this higher order ode into a system of first order ode's, we get: $$\vec{x}'=\vec{F}(x)$$ with $$ F_1(x)=...
1
vote
0answers
34 views

Finding a conjugation given a first integral

In the ODE given by: $x'=X(x)$ , where $X$ is my vectorial field $\in C^1$ in an open subset of $\mathbb R^n$ , If $X$ have $f$ a first integral and $df(p)\neq0$ then there exists a neighborhood $V_{...
-1
votes
2answers
79 views

Determine stability of non autonomous system at the origin

I am trying to determine the stability of the zero solution of the system $x'= \begin{bmatrix} -t & 1 \\ 1 & -t \end{bmatrix}x $ Even though, a Liapunov method can only ...
2
votes
1answer
130 views

Gradient systems and equilibrium points

I have been studying the following problem of gradient systems. Given a system: \begin{equation} \overset{\cdot}{x}=f(x), \ \ x=x(t) \in \mathbb{R}^3, \end{equation} where \begin{equation} f=-\nabla ...
1
vote
1answer
68 views

First Integral of 2 by 2 system

I have to show that in a linear 2 by 2 system if its a node, sink or source then it does not have first integrals. I tried considering a function $f(x,y)$, if its constant at a solution then $f(\phi)=...
1
vote
1answer
51 views

How to find stability of a third order non-linear system

Suppose we have a third order system, reduced to three first orders in the form $\dot x_1 = x_2 \\ \dot x_2 = x_1 + x_3F(x_1) \\ \dot x_3 = x_3F(x_1)$ Suppose we know $F(0) = 0$ How do we find the ...
0
votes
1answer
27 views

Finding a Poincaré map for the following ODE and discuss Stability.

I am trying to construct a Poincaré map for: $x’=p(t)x+q(t)$ Where $p(t) \ \text{and} \ q(t)$ are 1 periodic. I’m then asked to discuss the stability if, $\bar{p}=\int^{1}_{0}p(s)ds$. My first ...
1
vote
1answer
101 views

Implication of stability of Van der Pol oscillator.

Consider the homogeneous Van der Pol equation, $\ddot{x} + \mu (x^2-1)\dot{x} + x = 0$, with $\mu>0$. We convert it into a dynamical system, $$\dot{\bf x} = (y, -(x+\mu(x^2-1)y), \ \mathbf{x} \...
0
votes
0answers
15 views

Asymptotic stable Fixed point of function and its composition

I was trying to prove that Let $x^*$ be a fixed point of a continuous map f. Show that $x^*$ is asymptotically stable with respect to the map $g=f^2$, then it is asymptotically stable with respect to ...
4
votes
1answer
110 views

What's an example where Lyapunov fails to find the bounds of stability

In linear control theory, a system is stable if and only if if satisfies the Routh–Hurwitz stability criterion, so we can use this to solve for the limits of stability. E.g. you can find the maximum ...