Singularities & Limit Cycles of C1 Vector Fields on S2

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Discussion Overview

The discussion revolves around the properties of C1 vector fields on S2, specifically addressing the existence of singularities and the nature of isolated periodic orbits as limit cycles. The scope includes theoretical aspects of vector fields and their implications in dynamical systems.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants assert that any C1 vector field on S2 possesses at least one singularity, while others challenge this by suggesting that there are C1 vector fields on the torus without singularities, indicating a potential omission in the original claim.
  • One participant argues that since the orbit is periodic, it qualifies as a cycle, and due to its isolation, it must be classified as a limit cycle.
  • There is a clarification regarding the notation of S2, with one participant noting that S2 typically denotes a 2-sphere rather than a torus.
  • Another participant acknowledges the unconventional use of S2 to denote the torus, suggesting that while it is set-theoretically acceptable, it deviates from standard notation.

Areas of Agreement / Disagreement

Participants express disagreement regarding the existence of singularities in C1 vector fields on S2, with some asserting their presence and others denying it. The discussion about the classification of periodic orbits as limit cycles appears to have some consensus, but the notation issue remains a point of contention.

Contextual Notes

The discussion highlights potential limitations in definitions and notation, particularly regarding the interpretation of S2 and the conditions under which singularities may or may not exist in vector fields.

Who May Find This Useful

This discussion may be of interest to those studying dynamical systems, vector field theory, and mathematical notation in the context of topology and geometry.

johnson123
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(1) Show that any C1 vector Field on S2 (the torus) possesses at least one singularity.

(2)Show that any isolated periodic orbit T of a C1
planar vector field X is a limit cycle.

Any help/suggestions are appreciated.
 
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johnson123 said:
(1) Show that any C1 vector Field on S2 (the torus) possesses at least one singularity.

(2)Show that any isolated periodic orbit T of a C1
planar vector field X is a limit cycle.

Any help/suggestions are appreciated.


(1) There are C^1 vector fields on the torus without singularities. You must be omitting something.

(2) Since the orbit is periodic it is a cycle, and since it is isolated it must be a limit cycle.
 
S2 usually denotes a 2-sphere rather than a torus.
 
You're right... Johnson must have taken the liberty of denoting the Cartesian product [tex]S^2 := S\times S[/tex] for the torus, which is OK set-theoretically, but goes against standard notation.
 
Last edited:

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