We construct a symplectic atlas adapted to the flow action of an uncoupled isotropic n- oscillator, referred to as the Reeb atlas. In the context of Reeb's Theorem for Hamiltonian systems with symmetry, these variables are very useful for finding periodic orbits and determining their stability in perturbed harmonic oscillators. These variables separate orbits, meaning they are in bijective correspondence with the set of orbits. Hence, they are especially suited for determining the exact number of periodic solutions via reduction and averaging methods. Moreover, for an arbitrary polynomial perturbation, we provide lower and upper bounds for the number of periodic orbits according to the degree of the perturbation. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.