A study on multi-frequency patterns in nonlinear network oscillators using incremental harmonic balance method

Abstract This paper presents a study on multi-frequency pattern (MFP) vibration of nonlinear network systems based on the incremental harmonic balance (IHB) method. Instead of solving the considered highly-dimensional systems straightforwardly, the IHB method is proposed to solve a single-degree-of-freedom system with a time delay. The time delay is also incorporated into the iteration scheme, by introducing its relations with the phase lag and the angular frequency of limit cycle solutions. As the delay is adjusted to be in preset certain relations with the phase lag and the angular frequency, limit cycle solutions of the time-delayed system can be utilized to generate solutions of the original system. Two network systems are investigated to verify the proposed approach. The presented method can provide both stable and unstable limit cycles, and hence multiple solutions can be tracked successfully. Surprisingly, the coexistence of stable MFP is revealed in the van der Pol-type network oscillators. The attained results are useful for in-depth understanding of the mechanism of MFP. With such high efficiency and accuracy, moreover, the presented analysis method could be applicable in more nonlinear network systems.