Questions tagged [stability-in-odes]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
56 views

Why does this author choose the following?

Consider the following paper: https://reader.elsevier.com/reader/sd/pii/S1468121810003408?token=0705FEA1A105DA0B16C65CC64F443A6F68AF7DC9FC229E8C983B164443A63C29446280CF1284645D5EBCC33DA6610D5B&...
1
vote
1answer
35 views

Use the definition of stability of equilibrium to prove (1,0) is unstable. [duplicate]

Given the system $$x' = x(1-r)+y(x-r) \\ y' = y(1-r)+x(r-x)\\$$ where $r^2=x^2+y^2$. I don't know how to use the definition of stability of equilibrium to show that the equilibrium $(1,0)$ is ...
0
votes
0answers
37 views

Study instability of a system with Lyapunov functions in terms of a parameter

I am given the following system of odes \begin{gather} \begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix} - b(...
0
votes
0answers
19 views

How to check local stability of eigenvalues generated by a quintic characteristic polynomial?

Suppose we have a 5x5 matrix, how would we check the whether the eigenvalues are negative so that we can conclude they are locally stable? As requested in the comment by Woody3: here is the matrix: <...
1
vote
1answer
48 views

Trying to understand eigenvalues with respect to differential equations.

I am trying to understand how to find eigenvalues from a matrix consisting of exponential terms, considering a differential equation. The examples I've seen online are ODEs. Without using a vector ...
0
votes
0answers
16 views

Find an explicit example of a critical point $C$ said to be **Asymptotically stable** but not **stable**.

Is the critical point $C$ said to be stable and Asymptotically stable are same? I could find an example for a stable equilibrium point which is not asymptotically stable $$x'=-y$$ $$y'=x$$ I can ...
3
votes
1answer
57 views

Determine the stability of equilibrium point with Lyapunov function

I want to determine the stability of $(0,0)$ (stable, asymptotically stable or unstable) in the nonlinear system: $$ \begin{aligned} \dot{x} &= y + xy \\ \dot{y} &= -y + \sin^2(x)...
1
vote
0answers
42 views

Solving for Floquet multipliers: solving over multiple periods

Everywhere that I've read about performing a Floquet analysis involves numerically solving the system over one period, with the identity matrix as initial condition. My system is fairly complex, has ...
0
votes
1answer
101 views

How to show that $\Omega_{c}=\left\{x \in \mathbb R^{n}: V(x) \leq c\right\}$ is compact?

Consider an ODE system $$\dot{x}=f(x),$$ having a candidate Lypunov function, which satisfies $V(x)\geq0$, $V(0)=0$, and $\dot{V}(x)\leq0$. How to show that the set $$\Omega_{c}=\left\{x \in \mathbb R^...
0
votes
0answers
43 views

$\dot{V}(x)=0 \iff \dot{x}=0$ imply the system always converges to an equilibrium point?

Say we have an system of first-order ODE $$\dot{x}=\Phi(x), $$ which has several equilibrium point. We have found a positive definite Lyapunov function $V(x)$, in particular $V(x)$ is radially unbound ...
0
votes
0answers
25 views

Can locally Lipschitz ODE system imply bounded solutions?

Consider a system of first-order ODE $$\dot{x}=\Phi(x).$$ $\Phi(x)$ is locally Lipschitz. By the theorem, given any initial condition $x(t_0)=x_0$, the solution $x(t)$ is unique and continuous on ...
1
vote
0answers
107 views

System which escapes infinitely often

Is there a system of ordinary differential equations (over the reals) of the form $$ \begin{align} \dot{x}(t) &= f(x(t)) \\ x(0) &= x_0 \end{align} $$ with the following properties: $f(0) = 0$...
2
votes
0answers
61 views

References for Patterning Speed

Imagine I have a system describing two species concentrations $x$ and $y$ defined on a ring of $N$ cells, given by \begin{align} \dot{x}_j&=F_x(x_j,\{x_{r}\},y_j,\{y_{r}\})\\ \dot{y}_j&=F_y(...
2
votes
1answer
30 views

Stability of Differential Equations on the Argand Plane

My book (Advanced Engineering Mathematics 2nd Edition - MD Greenberg) states in Theorem 3.4.3 that in order for a system of Differential Equations to be stable "it is necessary and sufficient ...
1
vote
1answer
34 views

Poincare Map for Polar ODE

I am currently trying to obtain a Poincare Map for the ODE system originally given by \begin{cases}\dot{x} = (1-x^2-y^2)x-y\\\dot{y} = x+(1-x^2-y^2)y\end{cases} on the region $x \in (1/2, 3/2)$ and $y ...
2
votes
1answer
49 views

local stability of a planar autonomous system

I have been trying to prove that origin $(\theta,k) = 0$ of the non-autonomous system $$ \begin{aligned} \dot{\theta}(t) &= -b \theta(t) + k(t) u \\ \dot{k}(t) &= \epsilon \text{ } h(\theta(t),...
2
votes
1answer
47 views

Understanding where does the second (stochastic) attractor of the system come from.

I am currently reading a paper, studying the population dynamics in 3-dimensional Lotka-Volterra model with the following interaction change descriptive system: $$ \begin{equation} \begin{cases} ...
1
vote
1answer
118 views

Scalar Autonomous ODE: Stability of critical points using the derivative

Given a scalar autonomous differential equation of the form $\dot{x} = f(x)$. I want to investigate the stability of any critical point $x_0$. My textbook states that one can use the following ...
0
votes
1answer
8 views

Existence and uniqueness of solutions in the definition of stable equilibrium

So I'm reading Simulation Based Optimization: Parametric Optimization Techniques and Reinforcement Learning by Abhijit Gosavi. On page 312 there's a definition as follows: Definition 9.11. An ...
0
votes
0answers
31 views

How to find the equilibrium points and stability of a system of two diff. equations

I'm having a hard time working this out. My system discribes the evolution of the population of $2$ species: $$ \frac{dP}{dt} = (a - bP - kQ)P, \\ \frac{dQ}{dt} = (c - dQ - lP)Q. $$ As i understand, ...
3
votes
1answer
92 views

Stability analysis of non-linear system outside of equilibrium point

I have a non-linear system that comes from the transformation matrix for angular rotational rate from Euler -> Body frame: $$ \dot{r} = A + sin(r)tan(p)B + cos(r)tan(p)C \\ \dot{p} = cos(r) B - sin(...
1
vote
2answers
102 views

Stability of $\dot{x}=x^3-2xy,\dot{y}=x^2-y$

I was given the following exercise. Consider the nonlinear dynamical system $\dot{x}=x^3-2xy, \dot{y} = x^2-y$ (a) Find all fixed points. (b) Determine whether each fixed point is linearly stable or ...
0
votes
0answers
26 views

Perturbation around a steady state in 2 dimensions

I have been asked to investigate the stability of a steady state to the following equation: $\frac{\partial U}{\partial t} = D \frac{\partial^2 U}{\partial x^2} + rU (1- \frac{U}{K}) - EU$ with ...
1
vote
1answer
72 views

Prove $f^T(x)(\nabla^T f(x)+\nabla f(x))f(x)+x\cdot f(x) = 0$ if and only if $x=0$

Properties Given a $f:\mathbb R^n\to \mathbb R^n$. The function $f$ has the following properties $f(x) = 0 $ if and only if $x=0$. $\nabla f(x)$ are negative semidefine for any $x$. $f(x)$ is ...
1
vote
0answers
51 views

LaSalle's invariance principle with largest positive invariant set?

I have read many versions of LaSalle's invariance principle. One of them is the following: For $D\subset\mathbb{R}^{n}$ open let $f:D\to\mathbb{R}^{n}$ be $C^{1}(D;\mathbb{R}^{n})$. Consider $\dot{x}=...
1
vote
0answers
70 views

Understanding Asymptotic Fourier Behaviour

I've been struggling with understanding some key concepts on long term behaviour of Fourier modes. Imagine I have a $N\times M$ system of cells with two quantities being measured by $x_{i,j}(t)$ and $...
2
votes
1answer
55 views

Show existence of solutions to a vector field with any initial point on a Poincaré Map

There is a system of ODEs or vector field below: $$\dot x = x-y+x^2-x^3-xy^2$$ $$ \dot y =x+y+xy-x^2y-y^3$$ (a) let $\Sigma$ be the postive $x$ axis,$$\Sigma=\{(x,y)\in \mathbb R^2:x>0,y=0\}$$ Show ...
0
votes
1answer
34 views

How do you find the equilibrium points of this second order, non-linear ODE?

Here's the differential equation: $$\ddot{z}(t)=1-\frac{1}{(z(t)+z^\star)^2}\,u(t),$$ where $z^\star>0$ is a fixed parameter, and $u(t)$ is some arbitrary input. I know that at equilibrium, rates ...
2
votes
0answers
30 views

Routh-Hurwitz criterion not giving correct answer when done manually?

I asked this question on Mathematica(https://mathematica.stackexchange.com/questions/256213/routh-hurwitz-criterion-not-giving-correct-answer-when-done-manually) to get numerical results but I still ...
0
votes
1answer
40 views

Doubt about Lyapunov's theorem proof

Given the autonomous system $\dot x=f(x)$ and an equilibrium point $\bar x$, we know that it is stable if $\exists\phi:U_0\to \mathbb R$, $\phi\in\mathcal C^1(U_0;\mathbb R)$, with $ U_0$ open nbh of $...
1
vote
0answers
15 views

Lyapunov function of a transformed system

I have a system, which has the form $\frac{da_i}{dt}=\nu(\eta_i)(\sigma(\eta_i)-a_i)$, where $\eta=WA+V$ and $A,V,\eta\in\mathbb{R}^N$ are vectors and $W\in\mathbb{R}^{N\times N}$ is some square ...
4
votes
1answer
89 views

Does Asymptotic Stability Imply the Existence of a Lyapunov Function for a Nonlinear System?

For a linear time-invariant system $\dot x = Ax,$ the inverse Lyapunov theorem asserts that if the origin is asymptotically stable, then a Lyapunov function in the form $V(x) = x^\top P x$ for some ...
0
votes
1answer
31 views

Proof of asymptotic stability

The general solution of the homogeneous equation $Ly = 0$ is given by: $y(t) = c_1(t)e^{λ_1t} + c_2(t)e^{λ_2t} ... c_m(t)e^{λ_mt}$, where $c_j(t)$ is an arbitrary polynomial of degree $k_j − 1$. If $r$...
2
votes
1answer
95 views

Using $\delta$-$\varepsilon$ definition to prove stability for autonomous system

I want to prove that an equilibrium point of a simple autonomous system is stable using the $\delta$-$\varepsilon$ definition. By 'autonomous system' I mean a system that does not depend explicitly on ...
1
vote
0answers
38 views

can lyapunov exponent tell us the geometry of phase portrait?

Let's say we have a phase space that contains a global asymptotic attractor, thus the phase space would look like a basin, or say, a bowl. My question is how to analytically obtain the geometry of the ...
0
votes
1answer
46 views

What's the need of $\delta$ in the definition of Lyapunov stability?

I’m having trouble understanding Lyapunov stability, which is $\forall \epsilon > 0, \exists \delta > 0: \|\bar x(0)\| < \delta\implies \forall t \ge 0, \|\bar x(t)\| < \epsilon$ Why is ...
0
votes
0answers
42 views

does Lyapunov exponent depends on initial state?

Does Lyapunov exponent $\lim_{t\rightarrow\infty}\lim_{|\delta Z_0|\rightarrow 0}\frac{1}{t}\ln\frac{|\delta Z(t)|}{|\delta Z_0|}$ depends on initial state $Z_0$? I'm asking this question since I ...
2
votes
0answers
50 views

Can an unstable saddle point be asymptotically stable?

Consider a non-dimensional pendulum, $$ \frac{d^{2}}{d t^{2}} \theta{\left(t \right)} = - u \cos{\left(\theta{\left(t \right)} \right)} + \sin{\left(\theta{\left(t \right)} \right)}, $$ where $\theta$ ...
1
vote
0answers
33 views

A problem about stability of the origin in an ODE related to sign and parity of the order of the first non-zero $n$-th derivative

I am trying to solve this exam question Considering the scalar ODE $\dot x = f(x), x \in \mathbb{R}$, with $f \in C^{\infty}(\mathbb{R})$ and $f(0)=0 $and all the derivatives of $f $up to a certain ...
4
votes
1answer
129 views

Selecting a Lyapunov function for a SEI model

Consider the system: $$\frac{dS}{dt} =\nu N -\frac{\beta S I}{N} -\nu S$$ $$\frac{dE}{dt} =\frac{\beta S I}{N} -(\sigma+\nu)E $$ $$\frac{dI}{dt} = \sigma E -\nu I$$ where $N=S+E+I$. Removing ...
1
vote
1answer
95 views

Absolute stability of numerical methods for ODEs

I've troubles understanding the meaning of region of absolute stability for numerical methods for ODEs. I know that we can restric the study of stability of a certain method to the case of the test ...
2
votes
0answers
53 views

Linear Algebra Question and Matrix Perturbation Problem Solution

I found the following formulation of a perturbation problem in my research (see attached image). In step 4c to 4d, the equation goes from: $$|I+A^{-1}e_k\Gamma^T| = 0$$ $$|1+\Gamma^TA^{-1}e_k| = 0$$ ...
1
vote
0answers
40 views

Definition of globally uniformly asymptotically stable

In the book "Nonlinear systems" (Khalil, 3rd edition, page 150), an equilibrium is said to be globally uniformly asymptotically stable if it is uniformly stable, $\delta(\epsilon)$ can be ...
3
votes
0answers
59 views

Analysing global stability of a 1-dimensional system?

Consider \begin{align} \frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex] \frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I \end{align} Where $N=S+I$ is the total population. By substituting ...
2
votes
2answers
97 views

Derivative of a Lyapunov function for a nonlinear system

Let $$\begin{aligned}\begin{cases}\dot{x}_{1}=-\left( 2x_{1}-x_{2}\right)^3+\left( x_{1}-x_{2}\right) \\ \dot{x}_{2}= -\left( 2x_{1}-x_{2}\right) ^{3}+2\left( x_{1}-x_{2}\right)\end{cases}\\ \end{...
1
vote
1answer
97 views

Constant solutions of $\cfrac{dy}{dx} = \cfrac{x^{3}-x}{1+e^{x}}$

Consider the differential equation $\cfrac{dy}{dx} = \cfrac{x^{3}-x}{1+e^{x}}$. How to find constant solution $y(x)$ and $\displaystyle\lim_{x \to \infty}y(x),$ where $y(x)$ is the solution such that $...
1
vote
1answer
77 views

Finding the Lyapunov function

I have the following system of ODEs. \begin{equation} \dot{x} = -x + 4y \end{equation} \begin{equation} \dot{y} = -x - y^3 \end{equation} Typically, I have seen the first standard guess to be $V(x) = ...
-1
votes
1answer
35 views

Do local bifurcations possess global stability? [closed]

It is well-known from dynamical systems described by differential equations that the stability of fixed points is determined by employing the local stability analysis. However, to obtain a general ...
5
votes
0answers
220 views

Selecting a suitable Lyapunov function for the following systems to analyse global stability?

SECOND BOUNTY! i) SI MODEL Consider \begin{align} \frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex] \frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I \end{align} Where $N=S+I$ is the total ...
0
votes
1answer
55 views

How can I find the condition of all the stable and unstable equilibrium solutions of $y'=\sin(4y)$

Question: Find under what conditions of $n$ is $y=n\left(\frac{\pi }{4}\right)$ stable or unstable given $y'=\sin(4y)$. I actually have the solutions given by my lecturer, but there are still few ...

1
2 3 4 5
17