Questions tagged [limit-cycles]

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4
votes
2answers
237 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 $...
8
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0answers
448 views

A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)

Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
2
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1answer
99 views

Keeping track of limit cycles via certain second order differential operator

Inspired by the two posts which are linked bellow we ask the following question: Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...
2
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0answers
103 views

Irrational closed orbits of vector fields on $S^2$ (Limit cycles and trace formula)

Motivations: We first introduce our motivations: We wish to find an operator-theoretical interpretation for the number of limit cycles of a polynomial vector field on the plane. We quote the ...
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0answers
111 views

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 ...
6
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0answers
274 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 ...
4
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1answer
408 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?
5
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0answers
118 views

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 ...
3
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0answers
341 views

(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 $\gamma_1, \gamma_2$ ...
5
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0answers
67 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 ...
1
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1answer
367 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 ...
2
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0answers
66 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 ...
7
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2answers
577 views

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) \...
9
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2answers
226 views

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 ...
8
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1answer
328 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 ...
4
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2answers
156 views

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, ...
2
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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?
1
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0answers
44 views

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 ...
2
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0answers
214 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 ...
7
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0answers
481 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 ...
3
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1answer
276 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 $...
6
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2answers
1k views

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 ...
1
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1answer
88 views

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 ...
5
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1answer
360 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 ...
3
votes
1answer
192 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 ...
6
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0answers
513 views

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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0answers
112 views

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
117 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 ...
4
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1answer
255 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 ...
1
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1answer
210 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(...
2
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0answers
364 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 ...
37
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3answers
7k views

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 ...
1
vote
1answer
282 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 ...
6
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0answers
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) } \...
9
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2answers
541 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(...
2
votes
1answer
191 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 ...
3
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1answer
500 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 ...
8
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1answer
429 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 $M$ ...