A theory of modulational diffusion
An analytical theory of the phenomenon of ''modulational diffusion'' has been developed in a book by \textit{A. J. Lichtenberg} and the second author [Regular and stochastic motion (1983; Zbl 0506.70016)]. In the present paper this approach is continued and compared to numerical simulation. For this reason the authors consider the two degrees of freedom nonautonomous Hamiltonian \[ H(I_ 1,I_ 2,\theta_ 1,\theta_ 2)=\frac{I^ 2_ 1}{2}+\frac{I^ 2_ 2}{2}-\epsilon \cos (\theta_ 1+\lambda \sin \omega t)-\mu \cos (\theta_ 1-\theta_ 2). \] Here \(I_ i\) and \(\theta_ i\) denote action and angle variables; the first two terms describe unperturbed (integrable) motion of two oscillators, the third term introduces phase modulation of the first oscillator, whereas the last term couples the motion of both oscillators. If \(\epsilon \omega^{-2}\lambda^{-1/2}\) is sufficiently large (i.e., if the modulation frequency is sufficiently small) a modulational layer of overlapping resonances is produced, and the coupling to the second oscillator produces a long time diffusion along this layer. The diffusion rate for this process is evaluated analytically (within some approximations) and numerically. The diffusion coefficient is found to vary in a series of descending steps as the frequency difference between the two oscillators is increased.
scientific article; zbMATH DE number 3936040