Instability versus nonlinearity in certain nonautonomous oscillators: A critical dynamical t ransition driven by the initial energy

The equation of motion q" + Omega^2(t)q+|q|^(gamma-2)q=0 (gamma>2) for the real coordinate q(t) is studied, as an example of the interplay between nonlinearity and instability. Two contrasting mechanisms determine the behavior of q(t), when the time-varying frequency Omega(t) does produce exponential instability in the linear equation qlin"+Omega^2(t)qlin=0. At low energy, the exponential instability is the dominant effect, while at high energy the bounding effect of the autonomous nonlinear term prevails. Starting from low initial energies, the result of this competition is a time-varying energy characterized by quasiperiodic peaks, with an average recurrence time Tpeak. A closed critical curve S exists in the initial phase space, whose crossing corresponds to a divergence of the recurrence time Tpeak. The divergence of Tpeak has a universal character, expressed by a critical exponent a=1. The critical curve S is the initial locus of the solutions that vanish asymptotically. A close relationship exists between this dynamical transition and the transition from mobile to self-trapped polarons in one spatial dimension. The application to a number of physical problems is addressed, with special attention to the Fermi-Pasta-Ulam problem and to transitions to chaos. ©2001 The American Physical Society.