The GP map is a nonlinear generalization of the quantum kicked rotor , which has a nonlinearity included in the kick. It was introduced by Graham et al.. and it is known to exhibit superballistic growth of energy. I Numerically find that this phenomenon is independent of the presence of a kicking potential and is purely due to nonlinearity. Using Strichartz estimates I extablish rigorous quasi-exponential stability bounds. The rate of exponential growth of energy is just one Lyapunov exponent. Numerical analysis shows that orbits with a positive maximal Lyapunov exponent exist at any (positive) nonlinearity. For stationary orbits, the full spectrum of such exponents is analytically computable. The case when the kicking period is commensurate to 2\pi yields a perfectly well defined classical dynamical system , i.e. a measure preserving transformation on the unit sphere in a finite-dimensional Hilbert space, that undergoes a transition from integrability to ergodicity as the nonlinearity is increased.