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Questions tagged [limit-cycles]

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A complex limit cycle not intersecting the real plane(2)

Inspired by this question and the counter example provided in its answer we ask: Is there a polynomial vector field on $\mathbb{R}^2$ such that after complexification of the equation, the ...
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226 views

A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
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1answer
372 views

Updated background on Hilbert 16th problem?

What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
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Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle. According to this fact, we search for a related ...
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(Some possible obstructions to ) Limit cycles as closed geodesics(3)

First we explain our Motivation: Motivation: First note that there is no a Riemannian metric on an open set of the plane which possess two nested closed geodesics $\gamma_1, \gamma_2$ ...
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63 views

Numerical and computational approaches to limit cycle theory

I am searching for a big list of papers or researches devoted to limit cycles of planar polynomial vector fields based on a computational and numerical approach. I would like to ask ...
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1answer
309 views

A complex limit cycle not intersecting the real plane

Edit: This is a real coefficient version of the current post. Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow? There is ...
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63 views

A possible obstruction for existence of limit cycle for analytic vector field on $S^2$

Is there an analytic vector field $X$,on $S^2$ which possess a limit cycle but $X $, satisfy $\nabla_X X =0$ or satisfy $\nabla_X JX= 0$ where $J$ is the standard almost ...
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Canard limit cycle for certain singularly perturbed system(Is there a contradictory situation?)

From the figures of page 478 and 479 of this paper one find that the author probably means that we have a (canard) limit cycle for the system $$\begin{cases} x'=y-x^2\\ y'=\epsilon(a-x) ...
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2answers
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Is this Riccati equation (“Josephson junction”) always phase-locked at integer rotation numbers?

Given parameters $(a,k,A) \in \mathbb{R}^3$, we consider on $\mathbb{S}^1$ the $2\pi$-periodic ODE $$ \dot{\theta} \ = \ - a\sin(\theta) + k + A\cos(t) \hspace{4mm} \mathrm{mod} \ 2\pi. $$ Identifying ...
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1answer
305 views

Can a harmonic vector field possess a limit cycle?

Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)? Note that the Laplacian of a vector field is defined via natural correspondence ...
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2answers
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Polynomial vector field tangent to a given analytic simple closed curve

Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{R}^2$ which surrounds origin. Is there a polynomial vector field on the plane which is tangent to $\gamma$? In the other word, ...
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1answer
46 views

The number of limit cycles of a quadratic vector field with a unique singularity

Is there a uniform upper bound for the number of limit cycles of a quadratic vector field which has a unique singular point in the plane?
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Two semi stable limit cycles with disjoint interior

What is a precise example of a quadratic vector field on the plane with at least $1$ semi stable limit cycles? Furthermore, is there a quadratic polynomial vector field on the plane with two ...
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0answers
135 views

A cubic system with two nested limit cycles with opposite orientations(2)

The second part of Hilbert's 16th problem not only concerns "The number of limit cycles of a polynomial vector field", but also the position and configuration of of those limit cycles with respect to ...
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471 views

Limit cycles as closed geodesics(2)

Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the ...
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1answer
233 views

A cubic system with two nested limit cycles with opposite orientations

What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C_1,C_2$ of the system surrounds an annular region $R$ such that $...
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2answers
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The adjoint operators as elliptic operators

Edit: It seems that the link "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains a talk by Loic Teyssier about homological equations and vanishing cycles is temporally ...
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1answer
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Integral Separation of disjoint submanifolds of $\mathbb{R}^n$

Assume that $M_1, M_2, \ldots , M_k$ are $k$ disjoint compact submanifolds of $\mathbb{R}^n$ of the same dimension $m$. Assume that $\lambda_{ij}, \; 1\leq i,j\leq k$ are $k^2$ arbitrary ...
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1answer
346 views

Fredholm index vs. Limit cycle theory

Let $A$ be the algebra of all smooth functions $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is flat at the origin and is real analytic on $\mathbb{R}^2 \setminus \{0\}$. Let $B $ be ...
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1answer
160 views

When is a limit cycle generated by a Hamiltonian oval stable?

Consider a real polynomial $H$ of degree $n+1$ in the plane. A closed, connected component of a level curve $H=t$ is denoted by $\gamma(t)$ and called an oval of $H$. Let $\omega$ be a real 1-form ...
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Counting limit cycles via curvature in Riemannian geometry

In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem First we give a short introduction: A quadratic system is a polynomial vector field on ...
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A heat equation approach to the perturbation of vector field with center

Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it. We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=...
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0answers
113 views

An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...
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1answer
249 views

A different Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$

In this question $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ is the space of all smooth vector fields on the plane or sphere. A limit cycle for a vector field $X$ is an isolated closed ...
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1answer
207 views

Vector fields whose divergence are proper maps

Let $X$ be a polynomial vector field of degree $2$ on $\mathbb{R}^{2}$. Does there exist a nonvanishing smooth function $g$ such that $Div(gX)$ is a proper map?Or at least the zero locus of $Div(...
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294 views

Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
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2answers
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The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post.. For Mathematical ...
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1answer
274 views

Analytic vector fields on surfaces which have infinite number of singularities

Let $X$ be an analytic vector field on a compact oriantable surface $S$ with volume form $\omega$. We denote the set of its singularities by $Z(X)$. A local question Is there an analytic vector ...
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1k views

The Perturbation of Non Hamiltonian algebraic Vector fields

In this question we are interested in the number of limit cycles which appear in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } \...
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2answers
530 views

An algebraic Hamiltonian vector field with a finite number of periodic orbits(1)

Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question. Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x,y)+wQ(...
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1answer
182 views

The centralizer of Lienard equation

Consider the lienard vector field $\cases{ x'=y -F(x) \\ y'=-x } $ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
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1answer
473 views

limit cycles of dynamical systems

Consider $2D$ dynamical systems $X' = F(X)$ where $X=(x,y)$ is a 2-vector, and there is an equilibrium point at the origin. Let $L$ be the set of numbers $x > 0$ such that a limit cycle of the ...
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1answer
428 views

Existence of a vector field with a finite number of limit cycles.

The following question is related to the Seifert conjecture. Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...