Nonautonomous and nonlinear effects in generalized classical oscillators: A boundedness theorem

The phenomenon of exponential instability, arising in certain nonautonomous linear oscillators, motivates the question about the boundedness of amplitude and energy of the oscillators describing many physical situations. We present a rigorous result ensuring the boundedness for a class of generalized classical oscillators, characterized by symmetric potentials with only one equilibrium point. The key elements turn out to be the oscillating nature of the solutions and the presence of an autonomous part in the potential, diverging more than quadratically with the coordinate.