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

Tiago Carvalho; Luiz Fernando Gonçalves

doi:10.3934/dcdsb.2020287

Article Contents

2021, Volume 26, Issue 8: 4263-4280. Doi: 10.3934/dcdsb.2020287

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

Tiago Carvalho^1, and
Luiz Fernando Gonçalves^2, ,

1.
Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Avenida Bandeirantes, 3900, zip code 14040-901, Ribeirão Preto, SP, Brazil
2.
Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais, Rua São Luiz Gonzaga, zip code 35577-020, Formiga, MG, Brazil

^* Corresponding author: Luiz Fernando Gonçalves
^* Corresponding author: Luiz Fernando Gonçalves

Received: March 31, 2020

Revised: July 31, 2020

Early access: October 2020

Published: August 2021

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Abstract

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.

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

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Figure 1. Sliding Vector Field

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

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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 $.

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Figure 4. Trajectories in $ S^2 $

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Figure 5. Displacement function

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Figure 6. Trajectories of the vector field $ Z_1 $

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Figure 7. Trajectories of the vector field $ Z_2 $

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

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

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

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