Bifurcations and stability properties of nonlinear systems with symbolic software

Abstract

A sequence of symbolic algebra and algorithms, together with the corresponding software package, for the nonlinear analysis of vibrations, bifurcations and stability properties of autonomous and nonautonomous systems have been developed. The package has been designed, in particular, to handle complex phenomena in the vicinity of compound critical points. It also has the capacity to deal with multifrequency excitations in the framework of nonautonomous systems. The approach is based on a systematic perturbation technique, and it provides a convenient and advantageous tool for the nonlinear analysis of complex phenomena (e.g., high-dimensional tori). The algebra is presented in two distinct categories, depending on the properties of the Jacobian, characterizing the compound critical point of interest. Thus, the case in which all eigenvalues of the Jacobian are pure imaginary pairs has been treated separately from the case involving a combination of imaginary pairs and zero eigenvalues. Specifically, two examples have been analyzed—a forced system with two pairs of pure imaginary eigenvalues and a system with one pair and one zero eigenvalues under combined forcing and parametric excitation.