Windows of stability in control chaos

A three-parameter piecewise-linear difference model of a pulse-width-modulated feedback system is analyzed. Its dynamics are described for many parameter regimes, concentrating on the features that will be observable in real systems. In particular, explicit formulas are given for the (distinct) domains of existence of an infinite number of stable periodic solutions (of every period). Almost all initial conditions will converge to these solutions, so the observable behavior is very simple, even though coexisting with them are very many complicated, chaotic motions. It is argued that this is a common feature of nonlinear systems, so that in applications it will be necessary in general to gain knowledge of the structure of the system as a whole, in order to determine whether chaotic solutions (if any) are of physical significance.<
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