All Questions
26 questions
5
votes
1
answer
211
views
Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 807
2
votes
1
answer
153
views
What conditions on the rate matrix $Q$ ensure unique convergence in continuous-time Markov chains?
In the study of discrete-time Markov chains, the conditions under which all initial distributions converge to a unique stationary distribution are well-understood. Specifically, if the transition ...
- 807
1
vote
0
answers
108
views
Stability of rigid bodies spinning around $z$-axis under gravity
Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...
- 807
5
votes
1
answer
205
views
Bias of DS literature to polynomial ODEs
In the literature on continuous time dynamical system, we generally deal with an open set $U \subset \mathbb{R}^n$ and a vector field $F: U \rightarrow \mathbb{R}^n$ and define a DS by the ODE
$$\...
- 247
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 ...
1
vote
0
answers
99
views
Long-term behavior of asynchronous, stochastic, numerical solution to a dynamical system
I am simulating the behavior of a dynamical system, say $$\dot{x} = f(Ax; \lambda), $$
with an Euler update, where $x\in \mathbb{R}^n$ and $\lambda$ are some parameters. In my scenario, $A\in \mathbb{...
- 131
13
votes
1
answer
661
views
Poincaré on analytic dependence on parameters of solutions of linear differential equations
There is the following important General Principle: if a parameter enters
in a linear differential equation additively, for example
$$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$
where the parameter is $\...
- 91.8k
2
votes
0
answers
105
views
Bifurcations due to a nonlinearity parameter
Suppose we want to analyze the behavior of the system
$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+,
$$
...
- 243
2
votes
2
answers
296
views
Planar polynomial vector field for a harmonic pair of polynomials
Has the system of ODEs
$$\frac{dx}{dt}=P(x,y)\\
\frac{dy}{dt}=Q(x,y)
$$
been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of ...
- 7,127
7
votes
2
answers
259
views
Can a periodically additively perturbed sinusoidal vector field on the circle have a stable periodic orbit of higher least period?
I have heard that differential equations on $\mathbb{S}^1$ of the form
\begin{equation} \hspace{40mm} \dot{\theta}(t) \ = \ A\sin(\theta(t)) + g(t) \hspace{4mm} \mathrm{mod} \ 2\pi, \hspace{40mm} (1) \...
- 708
6
votes
0
answers
342
views
Had this theorem in Tresser's article been proven somewhere?
The article in question is About Some Theorems by L.P. Sil'nikov by Charles Tresser. I am interested in the theorem C from page 453 and a particular application of such theorem which is illustrated ...
- 165
2
votes
0
answers
226
views
Geometric ergodicity of dynamical system
I'm working with dynamical systems defined by ODEs and SDEs, in this latter case gradient systems in particular, a special case of Ito diffusions.
I've read that under reasonable assumptions this ...
- 205
6
votes
2
answers
862
views
A dynamical system defined by the Riemann zeta function
Let $\zeta$ be the classical Riemann zeta function.
We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector ...
- 356
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
1
vote
0
answers
70
views
What's best result for normal form theory of non-autonomous dynamical system [closed]
Given an autonomous dynamical system, we could find its rest points and then try to understand the grems of the vector field on the neighborhoods of the rest points. In particular, there is the famous ...
- 71
3
votes
0
answers
135
views
Motivation for the existence of periodic solutions [closed]
I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form
$$\ddot{...
- 41
2
votes
0
answers
280
views
Uniqueness of analytic center manifold
In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...
- 21
4
votes
0
answers
466
views
Lorenz attractor power spectrum
If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
- 41
1
vote
0
answers
243
views
A (different) foliation arising from Hopf fibration
In this question, first we fix an isomorphism between $TS^{3}$ and $S^{3}\times \mathbb{R}^{3}$.(To be more precise we consider the global trivialization of $TS^{3}$ with help of $3$ global ...
- 356
4
votes
1
answer
354
views
Reference request: Invariant sets of dynamical systems
(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ...
- 185
7
votes
2
answers
2k
views
The integral of torsion
I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced ...
- 356
3
votes
0
answers
193
views
Nonexistence of Limit Cycle
Consider a planar dynamical system described in polar coordinates as
$$
\left\{
\begin{array}{ll}
\dot{\theta}=\Delta - r \sin \theta,\\
\dot{r} = - r + 1 + \cos \theta,
\end{array}
\right.
$$
where $...
1
vote
1
answer
207
views
Number of solutions of a system of equation!
Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has ...
17
votes
5
answers
2k
views
2- and 3-body problems when gravity is not inverse-square
Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing
as $1/d^p$ for distance separation $d$ and some power $p$.
Two questions:
Presumably the 2-body ...
- 151k
4
votes
1
answer
1k
views
Limit of a discrete time dynamical system
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have ...
- 53
3
votes
1
answer
2k
views
A formula for the Jacobian of a flow
Let $U : \mathbb R^d \to \mathbb R^d$ be a smooth vector field, and let $F_t : \mathbb R \times \mathbb R^d \to \mathbb R^d$ be the corresponding smooth flow, defined by the differential equation $$\...
- 8,512