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Results about structural stability and the existence of limit cycles for piecewise smooth linear differential equations separated by the unit circle

  • Stability and Bifurcation - Memorial Issue Dedicated to Jorge Sotomayor
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Abstract

In this article, we investigate the structural stability and the existence of limit cycles in families of piecewise smooth differential equations where the unit circle serves as the discontinuity region. Our study encompasses families featuring singularities of center or saddle type, both visible and invisible, as well as those without any singularities. For the family that admits only constant vector fields, we describe the dynamics over \(S^1\) and present a result regarding structural stability. For the other families, we provide an upper bound for the number of limit cycles and present examples that illustrate the maximum number of limit cycles that can be realized. In the constant-center case, we present a proof of the existence and stability of the limit cycle using elementary analytical geometry. Additionally, we discuss the presence of homoclinic cycles in saddle-center cases for such differential equations, taking into account Filippov’s convention.

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Acknowledgements

The authors thank Espaço da Escrita - Pró-Reitoria de Pesquisa - UNICAMP - for the language services provided. We also thank Professor Jorge Sotomayor, our dear “Soto”, for his dedication to mathematics. Soto will be remembered not only for his high-level research but also for his mentorship of students and for your wise words to everyone who comes to you asking for advice. The field of the qualitative theory of dynamical systems will be forever grateful for his theorems. The moment for “questions or comments?” of the talks will never be the same without your sagacious, fun and not-easy-to-answer questions! Your academic family will forever cherish your memory.

Funding

Mayara Caldas was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001. Ricardo Martins was partially supported by FAPESP Grants 2021/08031-9, 2018/03338-6, CNPq Grants 315925/2021-3 and 434599/2018-2 and Unicamp/Faepex Grant 2475/21.

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  1. Departamento de Matemática, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, Campinas, SP, 13083–859, Brazil

    Mayara D. A. Caldas & Ricardo M. Martins

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  1. Mayara D. A. Caldas
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  2. Ricardo M. Martins
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Correspondence to Mayara D. A. Caldas.

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Communicated by Marco Antonio Teixeira.

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Caldas, M.D.A., Martins, R.M. Results about structural stability and the existence of limit cycles for piecewise smooth linear differential equations separated by the unit circle. São Paulo J. Math. Sci. (2024). https://doi.org/10.1007/s40863-024-00433-8

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