Application of the averaging method to the gyrokinetic plasma

we show that the solution to an oscillatory-singularly perturbed ordinary differential equation may be asymptotically expanded into a sum of oscillating terms. Each of those terms writes as an oscillating operator acting on the solution to a non oscillating ordinary differential equation with an oscillating correction added to it. The expression of the non oscillating ordinary differential equations are defined by a recurrence relation. We then apply this result to problems where charged particles are submitted to large magnetic field.