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

 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

Received  April 2020 Revised  August 2020 Published  October 2020

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.

Citation: Tiago Carvalho, Luiz Fernando Gonçalves. A flow on $S^2$ presenting the ball as its minimal set. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020287
References:
 [1] D. C. Braga, A. F. da Fonseca and L. F. Mello, Study of limit cycles in piecewise smooth perturbations of Hamiltonian centers via regularization method, Electronic Journal of Qualitative Theory of Differential Equations, 79 (2017), 1-13.  doi: 10.14232/ejqtde.2017.1.79.  Google Scholar [2] L. E. J. Brouwer, On continuous vector distributions on surfaces, in Proceedings of the Royal Netherlands Academy of Arts and Sciences (KNAW), 11 (1909), 850–858, https://www.dwc.knaw.nl/DL/publications/PU00013599.pdf. Google Scholar [3] C. A. Buzzi, T. de Carvalho and R. D. Euzébio, Chaotic planar piecewise smooth vector fields with non-trivial minimal sets, Ergodic Theory and Dynamical Systems, 36 (2016), 458-469.  doi: 10.1017/etds.2014.67.  Google Scholar [4] C. A. Buzzi, T. Carvalho and R. D. Euzébio, On Poincaré-Bendixson theorem and non-trivial minimal sets in planar nonsmooth vector fields, Publicacions Matemàtiques, 62 (2018), 113-131.  doi: 10.5565/PUBLMAT6211806.  Google Scholar [5] T. Carvalho and L. F. Gonçalves, Combing the hairy ball using a vector field without equilibria, Journal of Dynamical and Control Systems, 26 (2020), 233-242.  doi: 10.1007/s10883-019-09446-5.  Google Scholar [6] R. Cristiano, T. Carvalho, D. J. Tonon and D. J. Pagano, Hopf and Homoclinic bifurcations on the sliding vector field of switching systems in $\mathbb{R}^3$: A case study in power electronics, Physica D: Nonlinear Phenomena, 347 (2017), 12-20.  doi: 10.1016/j.physd.2017.02.005.  Google Scholar [7] T. Carvalho, D. D. Novaes and L. F. Gonçalves, Sliding Shilnikov connection in Filippov-type predator-prey model, Nonlinear Dynamics, 100 (2020), 2973-2987.   Google Scholar [8] T. de Carvalho, On the closing lemma for planar piecewise smooth vector fields, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1174-1185.  doi: 10.1016/j.matpur.2016.04.006.  Google Scholar [9] T. de Carvalho and D. J. Tonon, Generic bifurcations of planar Filippov systems via geometric singular perturbations, Bull. Belg. Math. Soc. Simon Stevin, 18 (2011), 861-881.   Google Scholar [10] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, Journal de Mathématiques Pures et Appliquées, 11 (1932), 333–376, http://eudml.org/doc/234887. Google Scholar [11] M. di Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations, International Journal of Bifurcation and Chaos, 11 (2001), 1121-1140.   Google Scholar [12] D. D. Dixon, Piecewise deterministic dynamics from the application of noise to singular equations of motion, Journal of Physics A: Mathematical and General, 28 (1995), 5539-5551.   Google Scholar [13] N. M. Drissa, Fixed Point, Game and Selection Theory: From the Hairy Ball Theorem to A Non Hair-Pulling Conversation, PhD thesis, Université Paris 1 Panthéon-Sorbonne, 2016, http://hdl.handle.net/10579/8840. Google Scholar [14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications, 1st edition, Springer Netherlands, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar [15] C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory and Dynamical Systems, 6 (1986), 17-44.  doi: 10.1017/S0143385700003278.  Google Scholar [16] A. Jacquemard and D. J. Tonon, Coupled systems of non-smooth differential equations, Bulletin des Sciences Mathématiques, 136 (2012), 239-255.  doi: 10.1016/j.bulsci.2012.01.006.  Google Scholar [17] T. Kousaka, T. Kido, T. Ueta, H. Kawakami and M. Abe, Analysis of border-collision bifurcation in a simple circuit, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353), 2 (2000), 481-484.   Google Scholar [18] V. Křivan, On the gause predator-prey model with a refuge: A fresh look at the history, Journal of Theoretical Biology, 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.  Google Scholar [19] R. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, 1st edition, Springer-Verlag Berlin Heidelberg, 2004. Google Scholar [20] J. Llibre, P. R. Silva and M. A. Teixeira, Regularization of discontinuous vector fields on $\mathbb{R}^3$ via singular perturbation, Journal of Dynamics and Differential Equations, 19 (2007), 309-331.   Google Scholar [21] J. Llibre and M. A. Teixeira, Regularization of discontinuous vector fields in dimension three, Discrete & Continuous Dynamical Systems - A, 3 (1997), 235-241.  doi: 10.3934/dcds.1997.3.235.  Google Scholar [22] J. Milnor, Analytic proofs of the "hairy ball theorem" and the brouwer fixed point theorem, The American Mathematical Monthly, 85 (1978), 521-524.  doi: 10.2307/2320860.  Google Scholar [23] L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 3rd edition, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar [24] S. H. Piltz, M. A. Porter and P. K. Maini, Prey switching with a linear preference trade-off, SIAM Journal on Applied Dynamical Systems, 13 (2014), 658-682.  doi: 10.1137/130910920.  Google Scholar [25] D. S. Rodrigues, P. F. A. Mancera, T. Carvalho and L. F. Gonçalves, Sliding mode control in a mathematical model to chemoimmunotherapy: The occurrence of typical singularities, Applied Mathematics and Computation, 387 (2020), 124782. doi: 10.1016/j.amc.2019.124782.  Google Scholar [26] F. D. Rossa and F. Dercole, Generic and generalized boundary operating points in piecewise-linear (discontinuous) control systems, in 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), (2012), 7714–7719. Google Scholar [27] A. J. Schwartz, A generalization of a Poincaré-Bendixson Theorem to closed two-dimensional manifolds, American Journal of Mathematics, 85 (1963), 453-458.   Google Scholar [28] P. A. Schweitzer, Counterexamples to the Seifert Conjecture and opening closed leaves of foliations, Annals of Mathematics, 100 (1974), 386-400.  doi: 10.2307/1971077.  Google Scholar [29] J. Sotomayor and A. L. F. Machado, Structurally stable discontinuous vector fields in the plane, Qualitative Theory of Dynamical Systems, 3 (2002), 227-250.  doi: 10.1007/BF02969339.  Google Scholar [30] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, in International Conference on Differential Equations, Lisboa, 1995, World Scientific Publishing, (1998), 207–223.  Google Scholar [31] E. T. Whittaker and G. Robinson, The Calculus of Observations: A Treatise on Numerical Mathematics, 4th edition, Blackie & Son limited, 1954. Google Scholar

show all references

References:
 [1] D. C. Braga, A. F. da Fonseca and L. F. Mello, Study of limit cycles in piecewise smooth perturbations of Hamiltonian centers via regularization method, Electronic Journal of Qualitative Theory of Differential Equations, 79 (2017), 1-13.  doi: 10.14232/ejqtde.2017.1.79.  Google Scholar [2] L. E. J. Brouwer, On continuous vector distributions on surfaces, in Proceedings of the Royal Netherlands Academy of Arts and Sciences (KNAW), 11 (1909), 850–858, https://www.dwc.knaw.nl/DL/publications/PU00013599.pdf. Google Scholar [3] C. A. Buzzi, T. de Carvalho and R. D. Euzébio, Chaotic planar piecewise smooth vector fields with non-trivial minimal sets, Ergodic Theory and Dynamical Systems, 36 (2016), 458-469.  doi: 10.1017/etds.2014.67.  Google Scholar [4] C. A. Buzzi, T. Carvalho and R. D. Euzébio, On Poincaré-Bendixson theorem and non-trivial minimal sets in planar nonsmooth vector fields, Publicacions Matemàtiques, 62 (2018), 113-131.  doi: 10.5565/PUBLMAT6211806.  Google Scholar [5] T. Carvalho and L. F. Gonçalves, Combing the hairy ball using a vector field without equilibria, Journal of Dynamical and Control Systems, 26 (2020), 233-242.  doi: 10.1007/s10883-019-09446-5.  Google Scholar [6] R. Cristiano, T. Carvalho, D. J. Tonon and D. J. Pagano, Hopf and Homoclinic bifurcations on the sliding vector field of switching systems in $\mathbb{R}^3$: A case study in power electronics, Physica D: Nonlinear Phenomena, 347 (2017), 12-20.  doi: 10.1016/j.physd.2017.02.005.  Google Scholar [7] T. Carvalho, D. D. Novaes and L. F. Gonçalves, Sliding Shilnikov connection in Filippov-type predator-prey model, Nonlinear Dynamics, 100 (2020), 2973-2987.   Google Scholar [8] T. de Carvalho, On the closing lemma for planar piecewise smooth vector fields, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1174-1185.  doi: 10.1016/j.matpur.2016.04.006.  Google Scholar [9] T. de Carvalho and D. J. Tonon, Generic bifurcations of planar Filippov systems via geometric singular perturbations, Bull. Belg. Math. Soc. Simon Stevin, 18 (2011), 861-881.   Google Scholar [10] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, Journal de Mathématiques Pures et Appliquées, 11 (1932), 333–376, http://eudml.org/doc/234887. Google Scholar [11] M. di Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations, International Journal of Bifurcation and Chaos, 11 (2001), 1121-1140.   Google Scholar [12] D. D. Dixon, Piecewise deterministic dynamics from the application of noise to singular equations of motion, Journal of Physics A: Mathematical and General, 28 (1995), 5539-5551.   Google Scholar [13] N. M. Drissa, Fixed Point, Game and Selection Theory: From the Hairy Ball Theorem to A Non Hair-Pulling Conversation, PhD thesis, Université Paris 1 Panthéon-Sorbonne, 2016, http://hdl.handle.net/10579/8840. Google Scholar [14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications, 1st edition, Springer Netherlands, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar [15] C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory and Dynamical Systems, 6 (1986), 17-44.  doi: 10.1017/S0143385700003278.  Google Scholar [16] A. Jacquemard and D. J. Tonon, Coupled systems of non-smooth differential equations, Bulletin des Sciences Mathématiques, 136 (2012), 239-255.  doi: 10.1016/j.bulsci.2012.01.006.  Google Scholar [17] T. Kousaka, T. Kido, T. Ueta, H. Kawakami and M. Abe, Analysis of border-collision bifurcation in a simple circuit, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353), 2 (2000), 481-484.   Google Scholar [18] V. Křivan, On the gause predator-prey model with a refuge: A fresh look at the history, Journal of Theoretical Biology, 274 (2011), 67-73.  doi: 10.1016/j.jtbi.2011.01.016.  Google Scholar [19] R. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, 1st edition, Springer-Verlag Berlin Heidelberg, 2004. Google Scholar [20] J. Llibre, P. R. Silva and M. A. Teixeira, Regularization of discontinuous vector fields on $\mathbb{R}^3$ via singular perturbation, Journal of Dynamics and Differential Equations, 19 (2007), 309-331.   Google Scholar [21] J. Llibre and M. A. Teixeira, Regularization of discontinuous vector fields in dimension three, Discrete & Continuous Dynamical Systems - A, 3 (1997), 235-241.  doi: 10.3934/dcds.1997.3.235.  Google Scholar [22] J. Milnor, Analytic proofs of the "hairy ball theorem" and the brouwer fixed point theorem, The American Mathematical Monthly, 85 (1978), 521-524.  doi: 10.2307/2320860.  Google Scholar [23] L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 3rd edition, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar [24] S. H. Piltz, M. A. Porter and P. K. Maini, Prey switching with a linear preference trade-off, SIAM Journal on Applied Dynamical Systems, 13 (2014), 658-682.  doi: 10.1137/130910920.  Google Scholar [25] D. S. Rodrigues, P. F. A. Mancera, T. Carvalho and L. F. Gonçalves, Sliding mode control in a mathematical model to chemoimmunotherapy: The occurrence of typical singularities, Applied Mathematics and Computation, 387 (2020), 124782. doi: 10.1016/j.amc.2019.124782.  Google Scholar [26] F. D. Rossa and F. Dercole, Generic and generalized boundary operating points in piecewise-linear (discontinuous) control systems, in 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), (2012), 7714–7719. Google Scholar [27] A. J. Schwartz, A generalization of a Poincaré-Bendixson Theorem to closed two-dimensional manifolds, American Journal of Mathematics, 85 (1963), 453-458.   Google Scholar [28] P. A. Schweitzer, Counterexamples to the Seifert Conjecture and opening closed leaves of foliations, Annals of Mathematics, 100 (1974), 386-400.  doi: 10.2307/1971077.  Google Scholar [29] J. Sotomayor and A. L. F. Machado, Structurally stable discontinuous vector fields in the plane, Qualitative Theory of Dynamical Systems, 3 (2002), 227-250.  doi: 10.1007/BF02969339.  Google Scholar [30] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, in International Conference on Differential Equations, Lisboa, 1995, World Scientific Publishing, (1998), 207–223.  Google Scholar [31] E. T. Whittaker and G. Robinson, The Calculus of Observations: A Treatise on Numerical Mathematics, 4th edition, Blackie & Son limited, 1954. Google Scholar
Sliding Vector Field
The $\omega$-limit of $p$ is disconnected
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 $\pi_S$.
Trajectories in $S^2$
Displacement function
Trajectories of the vector field $Z_1$
Trajectories of the vector field $Z_2$
Piecewise smooth vector field $Z_1$ and region $K_1$
Piecewise smooth vector field $Z_2$ and region $K_2$
Trajectory in $S^2$ passing through $p = (-1,0,0)$
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