Parametric resonance energy exchange and induction phenomenon in a one-dimensional nonlinear oscillator chain

We study analytically the induction phenomenon in the Fermi-Pasta-Ulam beta oscillator chain under initial conditions consisting of single mode excitation. Our study is based on the analytical computation of the largest characteristic exponent of an approximate version of the variational equation. The main results can be summarized as follows: (1) the energy density epsilon scaling of the induction time T is given by T approximately epsilon(-1), and T becomes smaller for higher-frequency mode excitation; (2) there is a threshold energy density epsilon(c) such that the induction time diverges when epsiloninfinity; (3) the threshold epsilon(c) vanishes as epsilon(c) approximately N-2 in the limit N-->infinity; (4) the threshold epsilon(c) does not depend on the mode number k that is excited in the initial condition; (5) the two modes k+/-m have the largest exponential growth rate, and m increases with increasing epsilon as m/N=sqrt[3betaepsilon]/pi. The above analytical results are thoroughly verified in numerical experiments. Moreover, we discuss the energy exchange process after the induction period in some energy density regimes, based on the numerical results.