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Results tagged with counterexamples
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user 36688
A counterexample is an example that disproves a mathematical conjecture or a purported theorem. For example, the Peterson graph is a counterexample to many seemingly plausible conjectures in Graph Theory.
3
votes
Results true in a dimension and false for higher dimensions
$S^{2}$ and $\mathbb{R}^{2}$ satisfies the Poincare Bendixon theorem but this theorem is not satisfied by higher dimensional spheres or Euclidean spaces.
For a related MSE post see the follo …
1
vote
2
answers
277
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Periodic orbit for certain vector field on $S^3$ (à la Seifert conjecture)
The standard frame for $S^3$ consists of $X_i,X_j,X_k$ with $X_i(a)=ia, X_j(a)=ja, X_k(a)=ka$ where $i,j,k$ are standard quaternion numbers, $a\in S^3$, and the multiplication is the quaternion multip …
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1
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1
answer
130
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A finite dimensional continuum with a subset $A$ such that both $A$ and $X\setminus A$ are d...
Inspired by this question we ask the following question.
Note that the comment conversation and answers to the above question imply that
There are two complementary subsets of the unit …
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2
votes
1
answer
494
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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 …
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2
votes
1
answer
131
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A special oscillatory orbit in space
Edit: According to the comment of Prof. Eremenko I revise the question.
19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact …
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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 …
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2
votes
0
answers
236
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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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2
votes
1
answer
210
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 cor …
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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
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