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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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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^{...
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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 ...
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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 ...
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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: $$ x'=0 \iff 2x^2-8a\lambda x +\lambda^2=0\tag{1} $$...
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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 ...
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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 $\...
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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 ...
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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^...
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Fixed Points and their Stability and Finally Bifurcation diagram

Find and Classify all fixed points and sketch vector field. What's the bifurcation? $$ 𝑥̇ = 𝑟𝑥 − \tan(𝑥)$$
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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 ...
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1answer
96 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 ...
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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 ...
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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 ...
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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}...
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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 ...
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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 ...
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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{$\ast$}\...
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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 $...
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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\...
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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 ...
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1answer
72 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 ...
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1answer
102 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 $f'\geq0$. ...
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1answer
88 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)=...
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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_{...
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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 ...
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1answer
117 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 ...
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1answer
53 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)=...
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1answer
42 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 ...
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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 ...
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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} \...
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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 ...
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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 ...
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Homogeneous ODEs of degree one

Consider a homogeneous ODE of degree one, i.e. $x'=f(x) \quad$ s.t. $\quad f(\alpha x)=\alpha f(x) \quad$ for $\alpha>0$ and $x\in \mathbb R^n$. Let $T_\alpha$ be the operator that multiplies a ...
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Does differentiating an integro-differential equation results in equivalent stability of the solution?

Consider the following integro-differential equation: $$\dot{x}(t)=ax(t)+b\int_0^tx(\tau)\text{d}\tau,$$ where $\dot{x}(t)$ denotes the time derivative of $x(t)$. If we derive the above equation and ...
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1answer
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Reason for choice of the word “asymptotically” stable in Lyapunov stability theory?

The equilibrium $x^\ast$ is (Lyapunov) stable iff $$\forall \varepsilon > 0 \; \exists \delta(\varepsilon) : \lvert x(0) - x^\ast \rvert < \delta(\varepsilon) \Rightarrow \forall t \geqslant 0 \;...
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Harvesting equation bifurcation

Could anyone give me some pointers on how to make a bifurcation diagram of a two parameter ODE of a harvesting model. It's $$ \dot{x} = ax\left(1 - \frac{x}{b}\right)- \frac{x^2}{1 + x^2}. $$ If I ...
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How to interpret Jacobi Stability?

I'm having the first contact with Jacobi stability for second order ODE, and I didn't understand very well what is the difference between the concept of Lyapunov stability nd Jacobi. It's very well ...
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1answer
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3rd Oder Homogeneous DE - stable/unstable equilibrium point

Consider a 3rd order linear homogeneous DE of the form $$Lu=u'''+a_2(x)u''+a_1(x)u'+a_0(x)u=f(x) \ \ \ \ (1)$$ and for which $u_1=e^{-x}$ and $u_2=e^{-2x}$ are solutions to the homogeneous form of $(...
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A question about stability

A dynamical system is governed by the equation $\frac{dx}{dt}=2\sqrt{1-x^2}$, $|x|\leq 1$. Then By equating $\frac{dx}{dt}$ to $0$ we get $1,-1$ are the fixed points. But how to check their stability? ...
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Prove that $0$ is locally exponentially stable

We consider the following O.D.E \begin{align}(a)\qquad\qquad\begin{cases}x'(t)=-x(t)+\left[\sin x(t)\right]^2 & t\geq 0,\\x(0)=x_0\in \Bbb{R}&\end{cases}\end{align} I want to prove that $0$ is ...
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How to determine the stability of the given polynomial?

Given a stable polynomial $\phi(s)=a_0+a_1{s}+a_2{s^2}+a_3{s^3}+\cdots+a_n{s^n}=\phi^{e}(s)+s\phi^{o}(s)$ where $\phi^{e}(s)=a_0+a_2{s^2}+a_4{s^4}+\cdots,~\phi^{o}(s)=a_1+a_3{s^2}+a_5{s^4}+\cdots$, ...
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Finding equilibrium points of a non-linear differential system

I'm trying to find the equilibrium points of $$ \begin{cases} \dot{x}=-1-y-e^{x} \\ \dot{y}=x^2+y(e^x-1) \\ \dot{z}=x+\sin(z) \\ \end{cases}$$ My try: $$\dot{x}=0\Rightarrow -1-y-e^x=0\Rightarrow y=...
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Stability of ODE involving trig functions and nonhyperbolic fixed points

Consider the following autonomous vector field: $$\dot x = −x$$ $$\dot y = \sin y$$ where $x \in \mathbb{R}^2, -\pi ≤ y ≤ \pi$ $\bullet$ Find all fixed points. $\bullet$ Determine the linearized ...
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1answer
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Stability of a 3D nonlinear ODE/dynamical system

I have tackled many 2D systems, but not any with 3D. I'm convinced that the principles and concepts still hold, however the problem is the lengthy computation of the eigenvalues as we have a 3x3 ...
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Stability of equilibrium points for a second order matrix ODE

Given an ODE $$\ddot{\mathbf{x}}=\mathbf{Ax}$$ where $\ddot{\mathbf{x}},\dot{\mathbf{x}}\in\mathbb{R}^n$ and $\mathbf{A}\in\mathbb{R}^{n\times n}$ is the symmetric matrix $$\begin{pmatrix}1&-0.5&...
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How to understand the Floquet Theory?

I am studying a physical case of a circular cylinder vibrating in a quiescent incompressible fluid. Such a system can be determined by two non-dimensional groups the Keulegan-Carpenter number and ...
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1answer
83 views

Lyapunov, asymptotic and BIBO stability of $4$ given systems

Which of the systems are Lyapunov, asymptotically or BIBO stable: $1) \quad \left[ \begin{array}{c|c} A & B\\ \hline C & \end{array} \right]=\left[ \begin{array}{ccc|c} -2&1&0&2\\ ...
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Center manifold of nonhyperbolic fixed point

Question: Consider the following autonomous vector field on the plane: $$\dot x = −x $$ $$\dot y = −x^2$$ $$(x,y) \in \mathbb{R}^2$$ $\bullet$ Compute the flow generated by this vector field. $\bullet$...
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How to describe a system with stability in terms of some variables

Assume I have a system: $\dot{\mathbf{x}}=-\frac{\partial}{\partial\mathbf{x}}L(\mathbf{x},\mathbf{y}) \\ \dot{\mathbf{y}}=\frac{\partial}{\partial\mathbf{y}}L(\mathbf{x},\mathbf{y})$ where $\...
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1answer
40 views

Positive definite Hessian implies Lyapunov stability

The Lagrange-Dirichlet Theorem is partially reversed by the following result: if the costraints are holonomous, bilateral, ideal, if $q^*$ is a critical point for the potential energy $U$, then the ...