10
votes
Hilbert 16th problem via hyperbolic geometry
This story is somewhat reminiscent of a (probably apocryphal) story of somebody writing a telegram to a mathematical research institute (Steklov Institute in Moscow, if I remember it correctly): "...
- 31.2k
10
votes
Accepted
A cubic system with two nested limit cycles with opposite orientations
It is not hard to concoct such an example in sufficiently high degree. For an example of degree $5$, take
$$
\begin{align}
x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\
y' &= y\,(1-x^2-y^...
- 108k
9
votes
Accepted
Updated background on Hilbert 16th problem?
An update from April 2018 is given by Patrick Speissegger.
The idea, going back to Poincaré, is to reduce the two-dimensional counting problem (counting limit cycles in the plane) to a one-...
- 188k
8
votes
The error in Petrovski and Landis' proof of the 16th Hilbert problem
• Q1 (the first yellow boxed question in the OP):
The current status of the "persistence problem" has been discussed by Ilyashenko in Complex length and persistence of limit cycles in a ...
- 188k
6
votes
What is the current status on methods to find limit cycles?
There are many principles to show the existence of periodic orbits in high- and infinite-dimensional systems, in particular, there are generalizations of the Poincaré-Bendixson theorem. I mention here ...
- 798
4
votes
Accepted
Can a harmonic vector field possess a limit cycle?
Consider the infinite strip, $[0,1]\times\mathbb{R}$ with $[0,y]$ identified with $[1,y]$ for all $y\in\mathbb{R}$, and with the Riemannian metric $g=dxdx+dydy$. Consider $X=\partial_x +f(y)\partial_y$...
- 156
3
votes
What is the current status on methods to find limit cycles?
This contains the question of the existence of periodic orbits (ot periodic solutions) in dynamical systems, a very wide question indeed. In dimensions >=3 it is much more complicated than in ...
- 2,479
3
votes
Polynomial vector field tangent to a given analytic simple closed curve
The answer is "no". In fact, it is still "no" for germs of curves : generically, a germ of an analytic curve $\gamma : (\mathbb R,0)\rightarrow (\mathbb R^2,0)$ is not tangent to any polynomial vector ...
- 5,428
3
votes
Accepted
Polynomial vector field tangent to a given analytic simple closed curve
I see this as very unlikely. A polynomial vector field would have a slope function that is a rational function of two variables with a finite number of coefficients.
As a consequence, if you take ...
- 5,186
2
votes
Accepted
The number of limit cycles of a quadratic vector field with a unique singularity
This survey seems to indicate that the answer is $1.$
- 96.4k
2
votes
The adjoint operators as elliptic operators
Consider the special case $M=\mathbb R^2$ and $X\equiv(1,0)$.
Now, given any vector field $Y$ in terms of component functions as $Y(x_1,x_2)=(y_1(x_1,x_2),y_2(x_1,x_2))$, a simple calculation gives
$$
...
- 8,086
2
votes
Accepted
Integral Separation of disjoint submanifolds of $\mathbb{R}^n$
In order for integration of differential forms to make sense, we need $M_1,\dotsc,M_k$ to be oriented. Let $\omega_i\in\Omega^m(M_i)$ be such that $\int_{M_i}\omega_i = 1$.
Since $M_1,\dotsc,M_k$ are ...
- 6,881
2
votes
A complex limit cycle not intersecting the real plane
A revision: Novembre 2020
I am realy indebted to Loic Teyssier for his $2$ very valuable comments and suggestions. I summarize his comments as follows:
To have a hyperbolic complex limit cycle ...
- 356
2
votes
Canard limit cycle for certain singularly perturbed system (Is there a contradictory situation?)
The figures 3.2 and 3.3 in Eckhaus's 1983 paper refer to the degenerate case that the function $f(x)$ in the differential equations
$$\begin{cases} x'=y-f(x)\\
y'=\epsilon(a-x) \end{cases}
$$
is ...
- 188k
2
votes
Accepted
A special kind of pseudo-garden eden states in cellular automata
Your definition is not very clear to me, but I'll try my best. Let $O \subset \{0, 1\}^*$ be the set of words (considered cyclic for the purpose of CA application) which lie on a limit cycle. Let $A \...
- 6,652
2
votes
Accepted
Mysteries of Wolfram's rule 18
Hopefully I guessed correctly that your two $1$s are next to each other, and your initial pattern is $110^{k-2}$ (considered periodically).
I'll do the "kink-elimination", to reduce the ...
- 6,652
2
votes
Accepted
Asymptotic behavior of system of differential equations
Preliminary remark. I am not certain whether Bendixon-Dulac grants the global attractiveness of the equilibria. However, via sheer leveraging on the (strict) monotonicity of $g$ and symmetry of the ...
- 767
2
votes
Asymptotic behavior of system of differential equations
The Bendixson–Dulac theorem is usually stated as you say, but in fact can be proven for arbitrary simply-connected regions (and generalized versions of it also hold for multiply-connected regions). ...
- 4,459
1
vote
Accepted
On proving the absence of limit cycles in a dynamical system
$$
\dot{X} = \dot{M} = \frac{1}{1+(E_0 + Y)^m } -a(M_0 +X) =
$$
$$
\frac{1}{1+E_0^m + mE_0^{m-1} Y + O(Y^2) } -aM_0 -aX =
$$
$$
\frac{1}{1+E_0^m} \left( 1-\frac{mE_0^{m-1} }{1+E_0^m } Y + O(Y^2) \...
- 6,696
1
vote
Accepted
Keeping track of limit cycles via certain second order differential operator
I believe the answer is yes.
Let $X$ be the vector field $2x \partial_y - y \partial_x$. The level sets of $y^2 + 2x^2$ are orbits of $X$, they have the shape of ellipses.
It is easy to compute $D(...
- 39k
1
vote
A complex limit cycle not intersecting the real plane(2)
This note contains an affirmative answer to the question
https://maco.lu.ac.ir/article-1-86-en.html
- 356
1
vote
Is this Riccati equation ("Josephson junction") always phase-locked at integer rotation numbers?
The answer to your question is YES, and you can look up the proof in our article with A. Klimenko https://arxiv.org/pdf/1305.6746.pdf for example.
The key idea is to mix monotonicity with Moebius ...
- 1,143
1
vote
The adjoint operators as elliptic operators
Any Lie bracket structure that is a first order differential operator is, by bilinearity and skew-symmetry, of the form
$$
\mathrm{ad}_X Y = A^{kp}_{ij}(X^i\partial_pY^j- Y^i\partial_pX^j)e_k,
$$
...
- 27.5k
1
vote
Fredholm index vs. Limit cycle theory
The linearization of the vector field $X$ at the singular point zero is
$$DX|_0 = \begin{pmatrix} 1 & 1\\ -1 & 0\end{pmatrix},$$
the eigenvalues of which are
$$ \lambda_{1, 2} = \frac{1}{2} \...
- 9,853
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