The primary and secondary instabilities of the sphere wake are investigated from the viewpoint of nonlinear dynamical systems theory. For the primary bifurcation, a theory of axisymmetry breaking by a regular bifurcation is given. The azimuthal spectral modes are shown to coincide with nonlinear modes of the instability, which provides a good reason for using the azimuthal expansion as an optimal spectral method. Thorough numerical testing of the implemented spectral–spectral-element discretization allows corroboration of existing data concerning the primary and secondary thresholds and gives their error estimates. The ideal axisymmetry of the numerical method makes it possible to confirm the theoretical conclusion concerning the arbitrariness of selection of the symmetry plane that arises. Investigation of computed azimuthal modes yields a simple explanation of the origin of the so-called bifid wake and shows at each Reynolds number the coexistence of a simple wake and a bifid wake zone of the steady non-axisymmetric regime. At the onset of the secondary instability, basic linear and nonlinear characteristics including the normalized Landau constant are given. The periodic regime is described as a limit cycle and the power of the time Fourier expansion is illustrated by reproducing experimental r.m.s. fluctuation charts of the streamwise velocity with only the fundamental and second harmonic modes. Each time–azimuthal mode is shown to behave like a propagating wave having a specific spatial signature. Their asymptotic, far-wake, phase velocities are the same but the waves keep a fingerprint of their passing through the near-wake region. The non-dimensionalized asymptotic phase velocity is close to that of an infinite cylinder wake. A reduced-accuracy discretization is shown to allow qualitatively satisfactory unsteady simulations at extremely low cost.