  We study the topological entropy of the magnetic flow on a closed riemannian
surface. We prove that if the magnetic flow has a non-hyperbolic closed orbit
in some energy set T^cM= E^{-1}(c), then there exists an exact $
C^\infty$-perturbation of the 2-form $ \Omega $ such that the new magnetic flow
has positive topological entropy in T^cM. We also prove that if the magnetic
flow has an infinite number of closed orbits in T^cM, then there exists an
exact C^1-perturbation of $ \Omega $ with positive topological entropy in T^cM.
The proof of the last result is based on an analog of Franks' lemma for
magnetic flows on surfaces, that is proven in this work, and Ma\~n\'e's
techniques on dominated splitting. As a consequence of those results, an exact
magnetic flow on S^2 in high energy levels admits a C^1-perturbation with
positive topological entropy. In the appendices we show that an exact magnetic
flow on the torus in high energy levels admits a $ C^\infty $-perturbation with
positive topological entropy.

Dynamical Systems