6.1: Introduction to Non-Linear Systems

The techniques discussed up to this point have all been developed for the analysis of linear systems. While the computational advantages of the assumption of linearity are legion, this assumption is often unrealistic, since virtually all physical systems are nonlinear when examined in suffi­cient detail. In addition to systems where the nonlinearity represents an undesired effect, there are many systems that are intentionally designed for or to exploit nonlinear performance characteristics.

Analytic difficulties arise because most of the methods we have learned are dependent on the principle of superposition, and nonlinear systems violate this condition. Time-domain methods such as convolution and fre­quency-domain methods based on transforms usually cannot be applied directly to nonlinear systems. Similarly, the blocks in a nonlinear block diagram cannot be shuffled with impunity. The absolute stability question may no longer have a binary answer, since nonlinear systems can be stable for certain classes of inputs and unstable for others.

The difficulty of effectively handling nonlinear differential equations is evidenced by the fact that the few equations we know how to solve are often named for the solvers. While considerable present and past research has been devoted to this area, it is clear that much work remains to be done. For many nonlinear systems the only methods that yield useful results in­volve experimental evaluation or machine computation.

This chapter describes two methods that can be used to determine the response or stability of certain types of nonlinear systems. The methods, while certainly not suited to the analysis of general nonlinear systems, are relatively easy to apply to many physical systems. Since they represent straightforward extensions of previously studied linear techniques, the in­sight characteristic of linear-system analysis is often retained.