All Questions
20 questions
5
votes
1
answer
212
views
Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 807
3
votes
0
answers
50
views
Stability of indefinitely damped mechanical system with diagonal stiffness
I'm trying to find conditions for the asymptotic stability of the following linear system,
\begin{equation}
\mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0
\end{equation}
given the ...
3
votes
1
answer
253
views
Stability analysis of equilibrium point of non-linear ODE system with Jacobian going to infinity
Let's say I have a nonlinear system of ODEs, where one of equations looks like:
$$
\frac{dX_i}{dt} = a_1X_0+\dotsb+a_j\frac{X_j^{0.5}}{X_j^{0.5}+b_j^{0.5}}+\dotsb.
$$
And equilibrium point is 0. I ...
- 31
1
vote
0
answers
53
views
Basin of attraction comparative statics* using local energy functions?
Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose ...
1
vote
1
answer
151
views
Analytically characterizing basins of attraction boundaries and sizes
While I understand that doing the above is not possible in general, I would like to know more about how to proceed when it is possible. That is, what are the common methods people use to analytically ...
2
votes
0
answers
153
views
Stability test for LTV systems by differential Lyapunov inequalities
Consider a linear time-varying system:
\begin{equation}
\dot x(t) = A(t) x(t), \tag{$*$}
\end{equation}
where $A(t)$ is a time-varying block matrix defined as
$$
A(t) =
\begin{bmatrix}
0 & I\\
-\...
- 55
1
vote
0
answers
62
views
Lyapunov theory in coupled nonlinear dynamic system with input
Suppose I have the following nonlinear coupled dynamic system
\begin{align*}
&\dot{x}_1 = f_1(x_1,x_2)\\
&\dot{x}_2 = f_2(x_2) + u
\end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{...
- 345
1
vote
0
answers
44
views
Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system
Consider the initial value problem
\begin{equation}\label{fainait ve}
\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; t \geq 0, \; \;\boldsymbol{f}(\boldsymbol{0}_n) = \boldsymbol{0}...
- 81
2
votes
1
answer
239
views
Seeking a Lyapunov function for a SIR model with immunity loss
We add the immunity loss to the SIR model and obtain the following autonomous system.
$$
\begin{align}
s' &= -is+\alpha r \\
i' &= i s - \gamma i\\
r' &= \gamma i-\alpha r
\end{align}
\...
- 2,239
8
votes
2
answers
1k
views
What is the current status on methods to find limit cycles?
What are the current best methods to show analytically the existence of a limit cycle in a $n$-dimensional system of the form:
$$
\frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x})
$$
Where $...
- 117
7
votes
1
answer
930
views
(In)stability of a two-dimensional dynamical system
Consider the following system of coupled differential equations
\begin{eqnarray*}
\dot{x}_1(t) & = & -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t)...
- 2,712
2
votes
1
answer
134
views
On local attractivity of a coupled non-linear differential equation
Consider a dynamical system described by the following coupled non-linear differential equation
\begin{align}
\dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\
\dot{x}_2(...
- 2,712
1
vote
0
answers
276
views
Stability when linearization fails
The dynamics of the $j$th system:
\begin{equation}
\begin{split}
\dot{\overline r}_j &= h (\overline r_j)
\,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...
- 33
2
votes
1
answer
65
views
Boundedness of particle motion with time-varying force
Consider the differential equation
$$ m \ddot{x} + k \dot{x} = - W_t x $$
where
$m$ and $k$ are nonnegative.
$x_t \in \mathbb{R}^n$
$W_t$ is a matrix that satisfies $$ \alpha I \succeq W_t \...
- 23
5
votes
0
answers
281
views
Basin of attraction of gradient flow
Suppose we have a compact Riemannian manifold $(M,g)$, and a Morse function $f : M \rightarrow \mathbb{R}$. Suppose we consider the gradient flow generated by this function, i.e. $$\dot{x_t} = - \...
- 151
3
votes
1
answer
202
views
Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
- 884
0
votes
1
answer
100
views
Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$
This question has been asked here but there is no answer:
https://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y
Consider autonomous ODE $y' = ...
- 103
3
votes
2
answers
430
views
Nonlinear ODE system: stability
I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...
- 31
0
votes
1
answer
285
views
Quadratic stability of linear time varying system
(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \...
- 1,590
0
votes
0
answers
320
views
Prove that origin is globally exponentially stable with Lyapunov Indirect Method
I'm wondering, if we have a nonlinear system governed by
$\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz
how can we show that the origin is globally exponentially stable?...
- 11