Multi-mode interactions in nonlinear circuits: validity limits of linear approximation

We discuss several types of potential instabilities in nonlinear electronic circuits due to sub- and ultra-harmonics generation, as well as intermodulation distortions caused by either multi-frequency input signals or coupling between intrinsic oscillatory modes. The limitations of traditional approach used for the circuit analysis in the time domain, like, e.g., harmonic balance or small-signal analysis approximation, are elucidated, and the constructive method for avoiding an unexpected destabilization of operation and thus providing a way of stable circuits design is proposed. Analytic criteria for stability are derived for a particular class of passive circuits composed of several lumped elements that is described by systems of ordinary differential equations with two or more degrees of freedom. The general condition predicting the appearance of chaotic oscillations, as well as other instabilities, such as saddle-note (flip) and sub-harmonic (period-doubling) bifurcations in multi-mode systems is obtained by means of the largest Lyapunov exponent analysis