Bifurcations associated with a three-fold zero eigenvalue

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Abstract
The bifurcation and instability behavior of a nonlinear autonomous system in the vicinity of a compound critical point is studied in detail. The critical point is characterized by a triple zero of index one eigenvalue, and the system is described by three independent parameters. The analysis is carried out via a unification technique, leading to a simple set of differential equations for the analysis of local behavior. Incipient and secondary bifurcations as well as bifurcations into invariant tori are discussed, and the explicit asymptotic results concerning periodic solutions are presented. Moreover, the criteria leading to a sequence of bifurcations into a family of two-dimensional tori are established. An electrical network is analyzed to illustrate the analytical results.

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