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A flow on $ S^2 $ presenting the ball as its minimal set

  • * Corresponding author: Luiz Fernando Gonçalves

    * Corresponding author: Luiz Fernando Gonçalves
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  • The main goal of this paper is to present the existence of a vector field tangent to the unit sphere $ S^2 $ such that $ S^2 $ itself is a minimal set. This is reached using a piecewise smooth (discontinuous) vector field and following the Filippov's convention on the switching manifold. As a consequence, none regularization process applied to the initial model can be topologically equivalent to it and we obtain a vector field tangent to $ S^2 $ without equilibria.

    Mathematics Subject Classification: Primary:34A36;34A12;37C10;37E35.


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  • Figure 1.  Sliding Vector Field

    Figure 2.  The $ \omega $-limit of $ p $ is disconnected

    Figure 3.  Item (a) shows the projection of the trajectories of $ X $ on $ S^2 $ by $ \pi_N $. In (b) we have the projection of the trajectories of $ Y $ by $ \pi_S $.

    Figure 4.  Trajectories in $ S^2 $

    Figure 5.  Displacement function

    Figure 6.  Trajectories of the vector field $ Z_1 $

    Figure 7.  Trajectories of the vector field $ Z_2 $

    Figure 8.  Piecewise smooth vector field $ Z_1 $ and region $ K_1 $

    Figure 9.  Piecewise smooth vector field $ Z_2 $ and region $ K_2 $

    Figure 10.  Trajectory in $ S^2 $ passing through $ p = (-1,0,0) $

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