# American Institute of Mathematical Sciences

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

## Periodic solutions of a periodic scalar piecewise ode

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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