On the subcritical period doubling of a non-smooth network system by incremental harmonic balance method

Abstract

This paper presents an investigation on an unusual bifurcation, i.e., subcritical period doubling (PD), of the non-smooth network system of a multi-modular floating structure. It is cumbersome to deduce the incremental harmonic balance iteration scheme, as non-smooth nonlinearities are resulted from the connectors between adjacent modules. To solve this problem, a method of undetermined coefficient is implemented plus a difference scheme. Periodic solutions are obtained accurately and a series of PDs are tracked as the controlling parameter varies. Interestingly, an unusual case is found to the Period-2 (with 2 times the fundamental period) solution, that is, the Period-4 solution appearing at the bifurcation point grows as the controlling parameter decreases backward. According to this feature, this case is called as subcritical PD compared to a traditional one. At the bifurcation point, the Period-2 solution loses stability featured by the leading Floquet multiplier leaving the unit cycle at +1, which is just opposite to −1 in the traditional cases. Differently, the arising Period-4 solution is unstable and it finally gains stability through cyclic bifurcations.