Reversible symmetric periodic solutions in spatial perturbed systems

Abstract We study the existence of several families of reversible-symmetric periodic solutions in a three-dimensional system of differential equations that admits a reversible symmetry and includes differentiable perturbations depending on a small parameter $\varepsilon $ , where the periodic solutions arise as continuations of circular solutions from the unperturbed system. We essentially impose symmetric constraints on the initial conditions and make use of the Poincaré continuation method. Both fixed-period and variable-period reversible-symmetric solutions are obtained. We provide sufficient conditions for their existence, expressed in terms of the perturbation functions. In addition, we compute the characteristic multipliers of these families of reversible-symmetric periodic solutions. We also compare different types of reversible symmetries with results from the averaging method. Several examples illustrating our results are presented.