Phase and average period of chaotic oscillators

Recently, there has been a great effort to extract the phase of chaotic attractors and complex oscillators. As a consequence many phases have been introduced, as example the standard phase θ based on the rotation of the vector position, and the phase ϕ based on the rotation of the tangent vector. Despite of the large interest in the phase dynamics of coupled oscillators there is still a lack of approaches that analyze whether these phase are equivalent and on what conditions these phases work. In this work, we show that the phase ϕ generalizes the standard phase θ, and it is equal to the length of the Gauss map, the generator of the curvature in differential geometry. Furthermore, we demonstrate, for a broad class of attractors, that the phase synchronization phenomenon between two coherent chaotic oscillators is invariant under the phase definition. Moreover, we discuss to which classes of oscillators the defined phases can be used to calculate quantities as the average frequency and the average period of oscillators. Finally, we generalize the phase ϕ which allows its use also to homoclinic attractors.