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A theory for the diagnosability of nonlinear dynamical systems, similar to the one in Part I  for memoryless systems, is developed. It is based on an input-output model of the system in a Hilbert space setting. A necessary and sufficient condition for the local diagnosability of the system, which is a rank test on a matrix, is derived. A simple sufficient condition is also derived. It is shown that, for locally diagnosable systems, there exist a finite number of test inputs that are sufficient to diagnose the system. Illustrative examples are presented.