Nonlinear Oscillations in Physical Systems

Abstract

Nonlinear oscillating systems are generally analyzed by approximation methods which involve some sort of linearization. These replace an actual nonlinear system with a so-called “equivalent” linear system and employ averaging which is not generally valid. While the linearizations commonly used are adequate in some cases, they may be grossly inadequate in others since essentially new phenomena can occur in nonlinear systems which cannot occur in linear systems. Thus, correct solution of a nonlinear system is much more significant a matter than simply getting more accuracy when we solve the nonlinear system rather than a linearized approximation. If we want to know how a physical system behaves, it is essential to retain the nonlinearity for complete understanding of behavior despite the convenience of linearity and superposition. Physical problems are nonlinear; linearity is a special case just as a deterministic system is a special case of a stochastic system. In a linear system, cause and effect are proportional. Such a linear relation sometimes occurs but is the exception rather than the rule. The general case is nonlinear and may be stochastic as well. In such cases, it is natural to make limiting assumptions—which is not always justified. Using decomposition, these become unnecessary even for the strongly nonlinear case and the case of stochastic (large fluctuation) behavior, as well as in the cases where perturbation would be applicable or in the linear and/or deterministic limits. “Smallness” assumptions, linearized models, or assumption of sometimes physically unrealistic processes may result, of course, in mathematical simplicity but again may not be justified in all circumstances.