The Nature and Use of Limit Cycles in Determining the Behavior of Certain Semideterminate Systems

In the context of this paper, a limit cycle is defined as the periodic output into which the behavior of a system is forced by controlling one or several periodic inputs. The conditions which lead to an operation under a limit cycle vary from system to system. A perfectly linear system may exhibit a level-insensitive, limit-cycle condition whenever the input frequencies bear integral relationship(s) with respect to each other. A highly nonlinear system may exhibit several level-sensitive limit-cycle operations for each integral relationship(s) between the input signal frequencies. General techniques for using these limit cycles to evaluate and categorize system behaviors are presented. However, it is up to the ingenuity of the experimentalist to use the techniques to obtain the information sought about the system. 1869 The coherence of the input signals is the fundamental concept behind the generation of limit cycles. For example, consider a network with several input ports. A set of input signals to the system would not produce a periodic response unless the inputs were mutually coherent, as well as individually periodic. The availability of stable frequency synthesizers and jitter-free digital signal generators has an immense effect on the stability of limit cycles and thus upon the design of experiments for evaluating the system behavior. In a sense, the technique is an extension of the single-input, single-output analysis which probes the system by changing the frequency and amplitude of the input signal.