Instability versus nonlinearity in certain nonautonomous oscillators: A critical dynamical transition driven by the initial energy - art. no. 026218

The equation of motion q + Omega (2)(t) q + alpha \q\(gamma -2) q = 0 (gamm a > 2) for the real coordinate q(t) is studied, as an example of the interp lay between nonlinearity and instability. Two contrasting mechanisms determ ine the behavior of q(t), when the time-varying frequency Omega (t) does pr oduce exponential instability in the linear equation q(lin) + Ohm (2)(t) q( lin) = 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 competiti on is a time-varying energy characterized by quasiperiodic peaks, with an a verage recurrence time T-peak. A closed critical curve S-omega exists in th e initial phase space, whose crossing corresponds to a divergence of the re currence time T-peak. The divergence of T-peak has a universal character, e xpressed by a critical exponent a = 1. The critical curve S-omega is the in itial locus of the solutions that vanish asymptotically. A close relationsh ip exists between this dynamical transition and the transition from mobile to self-trapped polarons in one spatial dimension. The application to a num ber of physical problems is addressed, with special attention to the Fermi- Pasta-Ulam problem and to transitions to chaos.