All Questions
Tagged with stability ca.classical-analysis-and-odes
22 questions
1
vote
0
answers
52
views
Stability of Euler discretization
I am looking at the discretization of an ODE:
$$x_{n+1} = x_n + \alpha f(x_n),$$
where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...
- 562
3
votes
2
answers
190
views
Floquet coefficients under time change
Let's consider two ODEs $\tag{1}\label{1}\frac{du}{dt}=\gamma(u(t))\ F(u(t))$ and $\tag{2}\label{2}\frac{dv}{d\tau}=F(v(\tau))$ where $f\in C^\infty(\mathbb R^n,\mathbb R^n)$ and $\gamma\in C^\infty(\...
- 41
3
votes
1
answer
159
views
Does gravity constant affect boundedness of solution?
Consider a second order gradient-like system with linear damping
$$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$
Suppose $f\in C^2(\mathbb{R}^n)$ and $\inf_{x\in\mathbb{R}^n}f(x)&...
- 311
5
votes
1
answer
284
views
Unbounded solution but bounded Euler discretization
Is there an ordinary differential equation in $\mathbb{R}^d$ induced by a gradient vector field with unbounded solutions, for which the difference equations obtained by using the forward Euler method ...
- 311
1
vote
1
answer
137
views
Stability of certain second order ODE
I am having a hard time determining the motion of $X$ in the ODE $X''+\nabla f(X)=0$ with initial conditions arbitrary $X(0)$ and zero velocity, i.e. $X'(0) = 0$. $X$ is in $\mathbb{R}^{n}$, and $f$ ...
- 11
2
votes
0
answers
153
views
Stability test for LTV systems by differential Lyapunov inequalities
Consider a linear time-varying system:
\begin{equation}
\dot x(t) = A(t) x(t), \tag{$*$}
\end{equation}
where $A(t)$ is a time-varying block matrix defined as
$$
A(t) =
\begin{bmatrix}
0 & I\\
-\...
- 55
1
vote
0
answers
157
views
Selecting a suitable Lyapunov function for the following systems?
i) SI MODEL
Consider
\begin{align}
\frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex]
\frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I
\end{align}
Where $N=S+I$ is the total population.
If ...
- 185
1
vote
0
answers
181
views
Subset of the domain of attraction
Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$
$$
\frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t))
$$
be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
- 61
4
votes
1
answer
118
views
almost linear ODE
Let $A,B$ be $n\times n$ matrices. I am interested in the following ODE in $\mathbb{R}^n$
$$ \frac{dx_t}{dt}=Ax_t+Bx^+_t $$
where $x_t^+=(x^+_{1,t},...,x^+_{n,t})$ and $(\cdot)^+$ is the rectifier: $...
- 315
3
votes
0
answers
82
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 ...
- 513
3
votes
1
answer
176
views
Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?
Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...
- 33
2
votes
1
answer
673
views
Center manifold theorem and case of all zero eigenvalues
Is the center manifold theorem applicable if say for a planar (2D) system of non-linear ODE, the stability matrix has both eigenvalues zero? Of course, there is only one eigenvector.
If not, what is ...
- 121
7
votes
1
answer
930
views
(In)stability of a two-dimensional dynamical system
Consider the following system of coupled differential equations
\begin{eqnarray*}
\dot{x}_1(t) & = & -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t)...
- 2,712
2
votes
1
answer
134
views
On local attractivity of a coupled non-linear differential equation
Consider a dynamical system described by the following coupled non-linear differential equation
\begin{align}
\dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\
\dot{x}_2(...
- 2,712
2
votes
0
answers
65
views
Exponential Convergence Under a Lyapunov-Like Assumption
Consider $V=\mathbb{R}^d\times\mathbb{R}^n$ with coordinate $x^T=[\theta^T,\sigma^T]$. I have an ODE of the form: $\dot{x}=F(x)$, where $F$ is assumed to be sufficiently smooth.
Suppose that there's ...
- 689
3
votes
1
answer
202
views
Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
- 884
5
votes
2
answers
828
views
Conditions for convergence to non-isolated fixed points
Consider a dynamical system of the form
$$
\dot{x}=f(x), \quad x\in X,
$$
and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov $V(x)$ ...
- 2,712
5
votes
0
answers
83
views
How to derive explicit bound for the solution of following equation?
Let's have equation
$$
y''(t) + \left(\frac{3}{16t^{2}} + \frac{a}{t} -\frac{b}{t^{\frac{5}{4}}}cos(2t)\right)y(t) = 0, \quad t \in (1, \infty), \quad a, b > 0
$$
How to derive explicit upper bound ...
- 161
2
votes
1
answer
116
views
A Statement from Brauer and Nohel's book on stability of time-depending linear systems
On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a $2 \times 2$ counterexample by Vinograd to the system $y'=A(t)y$, where
$$
A(t) =
\left(
\begin{...
- 21
0
votes
1
answer
285
views
Quadratic stability of linear time varying system
(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \...
- 1,590
1
vote
0
answers
176
views
Lyapunov stability, nonlinear system
Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article.
...
- 21
3
votes
1
answer
278
views
Lyapunov stability of linear system
Consider a linear ODE system
$$\dot x_k=\sum_{j=1}^ma_{kj}(t)x_j,\qquad k=1,\ldots, m,\quad a_{kj}(t)\in C[0,\infty).\quad (1)$$
Proposition. Suppose that $$\sup_{t\ge 0}\Big\{\int_0^t\Big(a_{kk}(...
- 31