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We present theoretical and numerical results pointing towards a strongconnection between the estimates for the diffusion rate along simple resonancesin multidimensional nonlinear Hamiltonian systems that can be obtained usingthe heuristic theory of Chirikov and a more formal one due to Nekhoroshev. Weshow that, despite a wide-spread impression, the two theories are complementaryrather than antagonist. Indeed, although Chirikov's 1979 review has thousandsof citations, almost all of them refer to topics such as the resonance overlapcriterion, fast diffusion, the Standard or Whisker Map, and not to theconstructive theory providing a formula to measure diffusion along a singleresonance. However, as will be demonstrated explicitly below, Chirikov'sformula provides values of the diffusion coefficient which are quite wellcomparable to the numerically computed ones, provided that it is implemented onthe so-called optimal normal form derived as in the analytic part ofNekhoroshev's theorem. On the other hand, Chirikov's formula yields unrealisticvalues of the diffusion coefficient, in particular for very small values of theperturbation, when used in the original Hamiltonian instead of the optimalnormal form. In the present paper, we take advantage of this complementarity inorder to obtain accurate theoretical predictions for the local value of thediffusion coefficient along a resonance in a specific 3DoF nearly integrableHamiltonian system. Besides, we compute numerically the diffusion coefficientand a full comparison of all estimates is made for ten values of theperturbation parameter, showing a very satisfactory agreement.

Chirikov and Nekhoroshev diffusion estimates: bridging the two sides of the river