The analysis of nonlinear systems driven by almost periodic inputs

We start in Chapter 1 by motivating the research, defining almost periodic and asymptotically almost periodic functions, reviewing known properties of these functions and introducing notation to be used in the other chapters.; In Chapter 2 we show how input-output stability theory bears on the problem of obtaining a frequency-domain stability criterion that can be used to design periodically driven varactor (nonlinear capacitor) circuits with guaranteed stability. We present an example of how a varactor frequency doubler with guaranteed stability may be designed using this criterion.; In Chapter 3 we give an analytical basis for evaluating the spectral coefficients for a large family of systems. This involves a convergent iterative process and certain bounds on the errors incurred in truncating the process.; In Chapter 4 we consider the equations of a large class of nonlinear circuits driven by asymptotically almost periodic inputs, and give an analytical basis for the use of harmonic balance to find steady-state solutions. More specifically, we show that in a certain setting there is a unique solution to the problem of obtaining a harmonic balance approximation, and that the approximations approach the actual solution as additional spectral components are included.; The results in Chapters 2–4 all involve a key circle-criterion hypothesis. In Chapter 5 we give an example that shows that this hypothesis cannot be relaxed significantly.; In Chapter 6 we give a generalization of the techniques in the first chapters that allows us to extend the theory to include non-diagonal nonlinearities. We give an example that shows the generalization is useful even in the case that the nonlinearity is diagonal.; In Chapter 7 we derive an integral equation that describes a nonlinear resistor in parallel with a nonlinear capacitor driven by a Thévenin equivalent source. We show how to obtain a contraction mapping operator on the set of square integrable functions and show the optimality of certain constants used in the construction of the contraction mapping operator.; We end in Chapter 8 by looking at an alternative method for determining the stability of steady state regimes in nonlinear circuits. We point out some problems in the theory and suggest a solution for one problem, but conclude that this method has some serious shortcomings.