Helicoidal trajectories of a charge in a nonconstant magnetic field

We investigate the existence of helicoidal trajectories for a charged particle in a magnetic field with fixed direction and whose intensity is of the form $b_{0}+\varepsilon b(p)$ where $b_{0}$ is a non zero constant, $b$ is some scalar function defined on the three-dimensional Euclidean space and $\varepsilon$ is a smallness parameter. We show that if $b$ depends just on one variable in a strictly monotone way then for every $\varepsilon\ne 0$ there is no simple helicoidal trajectory. On the other side, when the Poincar\'e-Melnikov function associated to the problem admits nondegenerate critical points or topologically stable extremal points then simple helicoidal trajectories exist and can be suitably localized. The existence results are based on the method of the finite-dimensional reduction.