Nonlinear Violence in Nonlinear Oscillations at High Energy

Abstract

This paper focuses on the characteristics of solutions of nonlinear oscillatory systems in the limit of very high oscillation energy, E; specifically, systems, in which the nonlinear driving force grows with energy much faster for x ( t ) close to the turning point, a(E) , than at any position, x(t) , that is not too close to a(E) . This behavior dominates important aspects of the solutions. It will be called “nonlinear violence”. In the vicinity of a turning point, the solution of a nonlinear oscillatory systems that is affected by nonlinear violence exhibits the characteristics of boundary-layer behavior (independently of whether the equation of motion of the system can or cannot be cast in the traditional form of a boundary-layer problem.): close to a(E) , x(t) varies very rapidly over a short time interval (which vanishes for E → ∞). In traditional boundary layer systems this would be called the “inner” solution. Outside this interval, x(t) soon evolves into a moderate profile (e.g. linear in time, or constant)—the “outer” solution. In (1 + 1)-dimensional nonlinear energy-conserving oscillators, if the solution is reflection-invariant, nonlinear violence determines the characteristics of the whole solution. For large families of nonlinear oscillatory systems, as E → ∞, the solutions for x(t) tend to common, indistinguishable profiles, such as periodic saw-tooth profiles or step-functions. If such profiles are observed experimentally in high-energy oscillations, it may be difficult to decipher the dynamical equations that govern the motion. The solution of motion in a central field with a non-zero angular momentum exhibits extremely fast rotation around a turning point that is affected by nonlinear violence. This provides an example for the possibility of interesting phenomena in (1 + 2)-dimensional oscillatory systems.