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September  2019, 24(9): 4851-4861. doi: 10.3934/dcdsb.2019034

Birth of an arbitrary number of T-singularities in 3D piecewise smooth vector fields

Tiago de Carvalho 1,, and  Bruno Freitas 2,

1. 

Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Av. Bandeirantes 3900, CEP 14040-901, Ribeirão Preto, SP, Brazil
2. 

Universidade Federal de Goiás, IME, CEP 74001-970, Caixa Postal 131, Goiânia, Goiás, Brazil

* Corresponding author: Tiago de Carvalho

Received  March 2018 Revised  July 2018 Published  September 2019 Early access  February 2019

Fund Project: The first author is partially supported by the CAPES grant number 1576689 (from the program PNPD) and also is grateful to the FAPESP/Brazil grants numbers 2013/34541-0 and 2017/00883-0, the CNPq-Brazil grant number 443302/2014-6 and the CAPES grant number 88881.030454/2013-01 (from the program CSF-PVE).

The T-singularity (invisible two-fold singularity) is one of the most intriguing objects in the study of 3D piecewise smooth vector fields. The occurrence of just one T-singularity already arouses the curiosity of experts in the area due to the wealth of behaviors that may arise in its neighborhood. In this work we show the birth of an arbitrary number, including infinite, of such singularities. Moreover, we are able to show the existence of an arbitrary number of limit cycles, hyperbolic or not, surrounding each one of these singularities.

Keywords: Periodic solutions, limit cycles, bifurcation, piecewise smooth vector fields.
Mathematics Subject Classification: Primary 34A36, 34A26, 37G15, 37G35.
Citation: Tiago de Carvalho, Bruno Freitas. Birth of an arbitrary number of T-singularities in 3D piecewise smooth vector fields. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4851-4861. doi: 10.3934/dcdsb.2019034
References:
[1]

T. Carvalho and B. R. Freitas, Birth of isolated nested cylinders and limit cycles in 3D piecewise smooth vector fields with symmetry, preprint, arXiv: 1702.01306(2016). Google Scholar

[2]

T. Carvalho and M. A. Teixeira, Attractivity, degeneracy and codimension of a typical singularity in 3D piecewise smooth vector fields, Preprint, arXiv: 1508.00456. Google Scholar

[3]

T. Carvalho and D. J. Tonon, Normal forms for codimension one planar piecewise smooth vector fields, International Journal of Bifurcation and Chaos, 24 (2014), 1450090 (11 pages). doi: 10.1142/S0218127414500904.  Google Scholar

[4]

A. Colombo and M. R. Jeffrey, The two-fold singularity of discontinuous vector fields, SIAM J. Appl. Dyn. Syst., 8 (2009), 624-640.  doi: 10.1137/08073113X.  Google Scholar

[5]

A Colombo and M. R. Jeffrey, Non-deterministic chaos, and the two fold singularity in piecewise smooth flows, SIAM J. Appl. Dyn. Syst., 10 (2011), 423-451.  doi: 10.1137/100801846.  Google Scholar

[6]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems - Theory and Applications, vol. 163 of Applied Mathematical Sciences. Springer-Verlag London, Ltd., London, 2008.  Google Scholar

[7]

M. di Bernardo, A. Colombo and E. Fossas, Two-fold singularity in nonsmooth electrical systems, Proc. IEEE International Symposium on Circuits ans Systems, (2011), 2713–2716. doi: 10.1109/ISCAS.2011.5938165.  Google Scholar

[8]

M. di Bernardo, A. Colombo, E. Fossas and M. R. Jeffrey, Teixeira singularities in 3D switched feedback control systems, Systems and Control Letters, 59 (2010), 615-622.  doi: 10.1016/j.sysconle.2010.07.006.  Google Scholar

[9]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers-Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[10]

M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, Journal of Differential Equations, 250 (2011), 1967-2023.  doi: 10.1016/j.jde.2010.11.016.  Google Scholar

[11]

A. Jacquemard, M. A. Teixeira and D. J. Tonon, Stability conditions in piecewise smooth dynamical systems at a two-fold singularity, Journal of Dynamical and Control Systems, 19 (2013), 47-67.  doi: 10.1007/s10883-013-9164-9.  Google Scholar

[12]

A. Jacquemard, M. A. Teixeira and D. J. Tonon, Piecewise smooth reversible dynamical systems at a two-fold singularity, International Journal of Bifurcation and Chaos, 22 (2012), 1250192, 13 pp. doi: 10.1142/S0218127412501921.  Google Scholar

[13]

A. C. J. Luo, Discontinuous Dynamical Systems, Springer, 2012. doi: 10.1007/978-3-642-22461-4.  Google Scholar

[14]

D. J. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science, Series A, 2010. doi: 10.1142/7612.  Google Scholar

[15]

J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 5-46.   Google Scholar

[16]

M. A. Teixeira, Stability conditions for discontinuous vector fields, J. Differential Equations, 88 (1990), 15-29.  doi: 10.1016/0022-0396(90)90106-Y.  Google Scholar

show all references

References:
[1]

T. Carvalho and B. R. Freitas, Birth of isolated nested cylinders and limit cycles in 3D piecewise smooth vector fields with symmetry, preprint, arXiv: 1702.01306(2016). Google Scholar

[2]

T. Carvalho and M. A. Teixeira, Attractivity, degeneracy and codimension of a typical singularity in 3D piecewise smooth vector fields, Preprint, arXiv: 1508.00456. Google Scholar

[3]

T. Carvalho and D. J. Tonon, Normal forms for codimension one planar piecewise smooth vector fields, International Journal of Bifurcation and Chaos, 24 (2014), 1450090 (11 pages). doi: 10.1142/S0218127414500904.  Google Scholar

[4]

A. Colombo and M. R. Jeffrey, The two-fold singularity of discontinuous vector fields, SIAM J. Appl. Dyn. Syst., 8 (2009), 624-640.  doi: 10.1137/08073113X.  Google Scholar

[5]

A Colombo and M. R. Jeffrey, Non-deterministic chaos, and the two fold singularity in piecewise smooth flows, SIAM J. Appl. Dyn. Syst., 10 (2011), 423-451.  doi: 10.1137/100801846.  Google Scholar

[6]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems - Theory and Applications, vol. 163 of Applied Mathematical Sciences. Springer-Verlag London, Ltd., London, 2008.  Google Scholar

[7]

M. di Bernardo, A. Colombo and E. Fossas, Two-fold singularity in nonsmooth electrical systems, Proc. IEEE International Symposium on Circuits ans Systems, (2011), 2713–2716. doi: 10.1109/ISCAS.2011.5938165.  Google Scholar

[8]

M. di Bernardo, A. Colombo, E. Fossas and M. R. Jeffrey, Teixeira singularities in 3D switched feedback control systems, Systems and Control Letters, 59 (2010), 615-622.  doi: 10.1016/j.sysconle.2010.07.006.  Google Scholar

[9]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers-Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[10]

M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, Journal of Differential Equations, 250 (2011), 1967-2023.  doi: 10.1016/j.jde.2010.11.016.  Google Scholar

[11]

A. Jacquemard, M. A. Teixeira and D. J. Tonon, Stability conditions in piecewise smooth dynamical systems at a two-fold singularity, Journal of Dynamical and Control Systems, 19 (2013), 47-67.  doi: 10.1007/s10883-013-9164-9.  Google Scholar

[12]

A. Jacquemard, M. A. Teixeira and D. J. Tonon, Piecewise smooth reversible dynamical systems at a two-fold singularity, International Journal of Bifurcation and Chaos, 22 (2012), 1250192, 13 pp. doi: 10.1142/S0218127412501921.  Google Scholar

[13]

A. C. J. Luo, Discontinuous Dynamical Systems, Springer, 2012. doi: 10.1007/978-3-642-22461-4.  Google Scholar

[14]

D. J. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science, Series A, 2010. doi: 10.1142/7612.  Google Scholar

[15]

J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 5-46.   Google Scholar

[16]

M. A. Teixeira, Stability conditions for discontinuous vector fields, J. Differential Equations, 88 (1990), 15-29.  doi: 10.1016/0022-0396(90)90106-Y.  Google Scholar

Figure 1.  Return map of $ Z = (X,Y) $
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Figure 3.  $ \Sigma- $centers at the T-singularities
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Figure 2.  Topological cylinders
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Figure 4.  Behavior of the orbits outside and inside of the planes
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Figure 5.  Behavior at Theorems B and C
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