Plenty of hyperbolicity on a class of linear homogeneous jerk differential equations

Abstract

We consider 3times 3 partially hyperbolic linear differential systems over an ergodic flow X^t and derived from the linear homogeneous differential equation dddot{x}(t)+beta (X^t(t))dot{x}(X^t(t))+gamma (t) x(t)=0. Assuming that the partial hyperbolic decomposition E^soplus E^coplus E^u is proper and displays a zero Lyapunov exponent along the central direction E^c we prove that some C^0 perturbation of the parameters beta (t) and gamma (t) can be done in order to obtain non-zero Lyapunov exponents and so a chaotic behaviour of the solution.