Periodic Solutions of a Class of Non-autonomous Discontinuous Second-Order Differential Equations

  • Clayton E. L. da SilvaEmail author
  • Alain Jacquemard
  • Marco A. Teixeira


We consider the second-order discontinuous differential equation y + η sgn(y) = 𝜃y + α sin(βt) where the parameters η, 𝜃, α, and β are real. The main goal is to discuss the existence of periodic solutions. Under explicit conditions, the number of such solutions is given. Furthermore, for each of these periodic solutions, an explicit formula is provided.


Non-smooth differential equations Second order Periodic solutions Non-autonomous Filippov systems 

Mathematics Subject Classification (2010)

34C15 34K12 34H05 



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Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto de Ciências Exatas e NaturaisUniversidade Federal de Mato Grosso (UFMT)RondonópolisBrazil
  2. 2.Sorbonne Universités, UPMC Univ Paris 06, and CNRS, UMR 7606ParisFrance
  3. 3.Inria, Paris Center, PolSys ProjectParisFrance
  4. 4.Institut de Mathématiques de Bourgogne, UMR CNRS 5584Université de Bourgogne Franche-ComtéFrance
  5. 5.Departamento de Matemática, Instituto de Matemática, Estatística e Computação CientíficaUniversidade de Campinas (UNICAMP)CampinasBrazil

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