On the periodic orbit bifurcating from a Hopf bifurcation in systems with two slow and one fast variables

The Hopf bifurcation in slow-fast systems with two slow variables and one fast variable has been studied recently, mainly from a numerical point of view. Our goal is to provide an analytic proof of the existence of the zero Hopf bifurcation exhibited for such systems, and to characterize the stability or instability of the periodic orbit which borns in such zero Hopf bifurcation. Our proofs use the averaging theory.The first and third authors are partially supported by a MICINN grant number MTM2011-22877 and by an AGAUR grant number 2009SGR 381. The second author is partially supported by a MICINN/FEDER grant number MTM2008- 03437, by an AGAUR grant number 2009SGR 410 and by ICREA Academia