American Institute of Mathematical Sciences

Advanced
March  2007, 6(1): 213-228. doi: 10.3934/cpaa.2007.6.213

Periodic solutions of a periodic scalar piecewise ode

José Luis Bravo 1, , Manuel Fernández 2, and  Antonio Tineo 3,

1. 

Departamento de Enseñanzas Básicas de la Ingeniería Naval, Universidad Politécnica de Madrid, Escuela Técnica Superior de Ingenieros Navales, 28040 Madrid, Spain
2. 

Departamento de Matemáticas, Universidad de Extremadura, Facultad de Ciencias, 06071 Badajoz, Spain
3. 

Departamento de Matemáticas, Universidad de Los Andes, Facultad de Ciencias, 5101 Mérida, Venezuela

Received  March 2006 Revised  September 2006 Published  December 2006

We study the distributions of periodic solutions of scalar piecewise equations defined by $x'=f(t,x)$ if $x\geq 0$, and $x'=g(t,x)$ if $x<0$, where $f,g$ are time periodic $\mathcal C^1$-functions such that $f(t,0)=g(t,0)$. Thus, these are equations on the cylinder where the vector field is not necessarily smooth on one of the equatorial circles.
We find that the solutions are $\mathcal C^1$-functions when the equation restricted to the equatorial line has a finite number of zeroes. Moreover, if $f$ and $g$ are analytic functions and the zeroes on the equatorial line are finite and simple, the set of periodic solutions consists of isolated periodic solutions and a finite number (determined by the number of zeroes) of closed "bands" of periodic solutions.
Keywords: Periodic solutions, scalar piecewise..
Mathematics Subject Classification: Primary: 34C25, Secondary: 34C0.
Citation: José Luis Bravo, Manuel Fernández, Antonio Tineo. Periodic solutions of a periodic scalar piecewise ode. Communications on Pure and Applied Analysis, 2007, 6 (1) : 213-228. doi: 10.3934/cpaa.2007.6.213
[1]

Evgeny Yu. Panov. On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws. Networks and Heterogeneous Media, 2016, 11 (2) : 349-367. doi: 10.3934/nhm.2016.11.349
[2]

Dag Lukkassen, Annette Meidell, Peter Wall. On the conjugate of periodic piecewise harmonic functions. Networks and Heterogeneous Media, 2008, 3 (3) : 633-646. doi: 10.3934/nhm.2008.3.633
[3]

Paola Buttazzoni, Alessandro Fonda. Periodic perturbations of scalar second order differential equations. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 451-455. doi: 10.3934/dcds.1997.3.451
[4]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[5]

Térence Bayen, Alain Rapaport, Fatima-Zahra Tani. Optimal periodic control for scalar dynamics under integral constraint on the input. Mathematical Control and Related Fields, 2020, 10 (3) : 547-571. doi: 10.3934/mcrf.2020010
[6]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[7]

Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59
[8]

A. Ducrot. Travelling wave solutions for a scalar age-structured equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 251-273. doi: 10.3934/dcdsb.2007.7.251
[9]

Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257
[10]

Shijin Deng, Weike Wang. Pointwise estimates of solutions for the multi-dimensional scalar conservation laws with relaxation. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1107-1138. doi: 10.3934/dcds.2011.30.1107
[11]

Wancheng Sheng, Tong Zhang. Structural stability of solutions to the Riemann problem for a scalar nonconvex CJ combustion model. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 651-667. doi: 10.3934/dcds.2009.25.651
[12]

Dingheng Pi. Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1055-1073. doi: 10.3934/dcdsb.2021080
[13]

Guo Lin, Shuxia Pan. Periodic traveling wave solutions of periodic integrodifference systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3005-3031. doi: 10.3934/dcdsb.2020049
[14]

Xiaoyan Lin, Xianhua Tang. Solutions of nonlinear periodic Dirac equations with periodic potentials. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2051-2061. doi: 10.3934/dcdss.2019132
[15]

Claudianor O. Alves, Giovany M. Figueiredo, Gaetano Siciliano. Ground state solutions for fractional scalar field equations under a general critical nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2199-2215. doi: 10.3934/cpaa.2019099
[16]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729
[17]

David L. Finn. Noncompact manifolds with constant negative scalar curvature and singular solutions to semihnear elliptic equations. Conference Publications, 1998, 1998 (Special) : 262-275. doi: 10.3934/proc.1998.1998.262
[18]

Y. Kabeya. Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 117-134. doi: 10.3934/dcds.2006.14.117
[19]

Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917
[20]

Leszek Gasiński. Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 143-158. doi: 10.3934/dcds.2007.17.143

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy