# Questions tagged [limit-cycles]

The limit-cycles tag has no usage guidance.

51
questions

3
votes

0
answers

193
views

### Jacobi equation and conjugate points on solution curves of the Van der Pol vector field

Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...

0
votes

1
answer

106
views

### A special kind of pseudo-garden eden states in cellular automata

I'm currently investigating Wolfram's elementary cellular automata on finite grids with periodic boundary conditions, i.e. on $\mathbb{Z}/k$ for different $k$.
It is clear that for each rule $R$ and ...

2
votes

1
answer

366
views

### Mysteries of Wolfram's rule 18

[Unfortunately, I made some mistakes in my original question. I tacitly corrected them wherever I found them.]
Wolfram's rule 18 gives rise to fractal patterns, but when started with two black cells ...

0
votes

0
answers

96
views

### Possible shifts in finite elementary cellular automata

I investigated the long term behaviour of a pair of black cells ■■ on a circle of $N$ cells under the action of each of Wolfram's rules $R$. For each combination $(R,N)$ I determined the first ...

1
vote

0
answers

362
views

### Astonishing affinity of Wolfram's rule 110 to the numbers 2 and 7

I investigated the evolution of a single black cell on 1-dimensional grids with periodic boundary conditions of variable sizes $N$ under Wolfram's rule 110 which is the only one for which Turing ...

1
vote

0
answers

95
views

### Limit cycles or stable solutions for k-dimensional piece-wise linear ODEs

As a branch of reinforcement learning, restless multi-armed bandits have been shown PSPACE-HARD but Whittle has offered an implementable solution called the Whittle Index Policy. Weber and Weiss ...

2
votes

2
answers

297
views

### Asymptotic behavior of system of differential equations

Let $f:[0,1]^2 \rightarrow \mathbb{R}^2,$ where $f_1(x,y) = g(y)-x$ and $f_2(x,y) = g(x)-y.$ Here $g(\cdot)$ is a strictly decreasing polynomial function such that $g(0)=1$ and $g(1)=0.$ I am ...

3
votes

0
answers

105
views

### An algebraic foliation of $\mathbb{C}^2$ which admits a non-algebraic complex limit cycle $L$ with $L\cap \mathbb{R}^2=\emptyset$

Is there a polynomial vector field on $\mathbb{R}^2$ whose corresponding singular complex foliation of $\mathbb{C}^2 $ admits a complex limit cycle $L$ such that $L$ does not intersect the real ...

4
votes

0
answers

142
views

### An algebraic foliation of $\mathbb{C}^2$ with real coefficients whose all complex limit cycles intersect the real plane $\mathbb{R}^2$

Is there a polynomial vector field $X$ on $\mathbb{R}^2$ such that every complex limit cycle
of the corresponding foliation of $\mathbb{C}^2$ must necessarily intersect the real plane $\mathbb{R}^2$.
...

1
vote

0
answers

61
views

### Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?

Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties?
The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...

0
votes

1
answer

113
views

### On proving the absence of limit cycles in a dynamical system

I'm studying this classical paper in nonlinear dynamics in biophysics and I want to understand it properly. I'm stuck at this two-variable problem which I've struggled with for too long now.
$$ \dot M ...

6
votes

2
answers

941
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
votes

0
answers

491
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
votes

1
answer

119
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
votes

0
answers

123
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 ...

2
votes

1
answer

209
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
votes

0
answers

282
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
votes

1
answer

553
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
votes

0
answers

137
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
votes

0
answers

355
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
votes

0
answers

76
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 ...

2
votes

1
answer

491
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 a ...

2
votes

0
answers

69
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
votes

2
answers

627
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
votes

2
answers

297
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
votes

1
answer

363
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
votes

2
answers

186
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
votes

1
answer

51
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?

7
votes

1
answer

819
views

### Hilbert 16th problem via hyperbolic geometry

More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a ...

3
votes

0
answers

132
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 one semi stable limit cycles?
Furthermore, is there a quadratic polynomial vector field on the plane with two ...

2
votes

0
answers

234
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
votes

0
answers

510
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 ...

4
votes

0
answers

476
views

### Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)

This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow.
Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...

4
votes

1
answer

350
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
votes

2
answers

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
vote

1
answer

95
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
votes

1
answer

407
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

1
answer

237
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
votes

0
answers

534
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 ...

0
votes

0
answers

121
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)=...

6
votes

0
answers

433
views

### An algebraic 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
votes

1
answer

281
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
vote

1
answer

229
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
votes

0
answers

412
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 ...

38
votes

3
answers

8k
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 development ...

1
vote

1
answer

338
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 ...

14
votes

1
answer

1k
views

### The perturbation of non-Hamiltonian algebraic vector fields

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

9
votes

2
answers

622
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

1
answer

211
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
votes

1
answer

525
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 ...