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

Periodic solutions of a periodic scalar piecewise ode


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


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


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.
Citation: José Luis Bravo, Manuel Fernández, Antonio Tineo. Periodic solutions of a periodic scalar piecewise ode. Communications on Pure & Applied Analysis, 2007, 6 (1) : 213-228. doi: 10.3934/cpaa.2007.6.213

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