Singularities & Limit Cycles of C1 Vector Fields on S2

In summary, any C1 vector field on S2 (the torus) will have at least one singularity, and any isolated periodic orbit of a C1 planar vector field X is a limit cycle. These conclusions have been proven, but it should be noted that the use of S2 to denote a torus goes against standard notation.
  • #1
johnson123
17
0
(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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  • #2
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.
 
  • #3
S2 usually denotes a 2-sphere rather than a torus.
 
  • #4
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:

1. What are singularities and limit cycles in the context of C1 vector fields on S2?

Singularities refer to points on a vector field where the vector is either zero or undefined. These points can be classified as either sources, sinks, saddles, or centers. Limit cycles, on the other hand, are closed trajectories on the vector field that do not intersect with themselves. They can occur around singularities or in other regions of the vector field.

2. How are singularities and limit cycles related in C1 vector fields on S2?

Singularities and limit cycles are closely related in C1 vector fields on S2. In fact, limit cycles can often be found around singularities, as they represent stable or unstable equilibrium points in the vector field. The type of singularity also determines the type of limit cycle that can occur.

3. What are the main properties of singularities and limit cycles in C1 vector fields on S2?

The properties of singularities and limit cycles in C1 vector fields on S2 include stability, index, and orientation. Stability refers to whether a singularity is stable (attracts nearby trajectories) or unstable (repels nearby trajectories). The index of a singularity is a measure of how many times the vector field winds around it. Orientation refers to the direction in which the vector field rotates around the singularity.

4. How are singularities and limit cycles visualized in C1 vector fields on S2?

Singularities and limit cycles can be visualized using phase portraits, which show the trajectories of the vector field in a specific region. Singularities are indicated by the type of point (source, sink, etc.), while limit cycles appear as closed curves. These visualizations can provide insights into the behavior of the vector field near singularities and limit cycles.

5. What are some applications of studying singularities and limit cycles in C1 vector fields on S2?

Studying singularities and limit cycles in C1 vector fields on S2 has various applications in mathematics and physics. For example, it can be used to understand the behavior of dynamical systems, model biological or ecological processes, and analyze the stability of physical systems. This knowledge can also be applied in engineering, such as in the design of control systems or the study of fluid dynamics.

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