Skip to main content

All Questions

Filter by
Sorted by
Tagged with
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) \...
Zhang Yuhan's user avatar
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 ...
Zhang Yuhan's user avatar
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 ...
Zhang Yuhan's user avatar
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 $$\...
NicAG's user avatar
  • 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 ...
cluelessmathematician's user avatar
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{...
MRicci's user avatar
  • 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 $\...
Alexandre Eremenko's user avatar
2 votes
0 answers
105 views

Bifurcations due to a nonlinearity parameter

Suppose we want to analyze the behavior of the system $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+, $$ ...
garserdt216's user avatar
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 ...
M.G.'s user avatar
  • 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) \...
Julian Newman's user avatar
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 ...
Evgeny's user avatar
  • 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 ...
nabla's user avatar
  • 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 ...
Ali Taghavi's user avatar
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 $\...
valle's user avatar
  • 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 ...
user20165678's user avatar
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{...
digiboy1's user avatar
2 votes
0 answers
281 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 ...
aristote's user avatar
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 ...
Basil's user avatar
  • 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 ...
Ali Taghavi's user avatar
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 ...
ttb's user avatar
  • 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 ...
Ali Taghavi's user avatar
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 $...
Mohammad Khosravi's user avatar
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 ...
Mohammad Khosravi's user avatar
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 ...
Joseph O'Rourke's user avatar
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 ...
Eric Blair's user avatar
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 $$\...
Tom LaGatta's user avatar
  • 8,512