Unveiling Pseudo-Period-Tripling Bifurcations in Nonlinear Dynamical Systems

This study aims to precisely analyze period-tripling bifurcations reported in the literature, specifically addressing the challenge of distinguishing between genuine and pseudo-period-tripling bifurcations caused by numerical errors or minor parameter variations. Current research on period-tripling bifurcations is limited, with most studies relying on numerical methods for solutions and identifying period-tripling bifurcations by observing numerical bifurcation diagrams. However, these methods may lead to misjudgments due to numerical and visual inaccuracies. In order to address this issue, this study employs the incremental harmonic balance method for accurate solutions and quantitative analysis of typical nonlinear systems. The Floquet theory, phase diagrams, frequency spectra, bifurcation diagrams, and Poincaré sections are used to analyze two types of pseudo-period-tripling bifurcations meticulously. The results indicate that certain observed period-tripling bifurcation phenomena (e.g. bifurcations from stable period-1 to unstable period-3 and from stable period-2 to stable period-6) are actually pseudo-bifurcations caused by numerical errors or visual misinterpretations rather than real dynamic behavior changes. This study provides a more accurate and reliable method to investigate period-tripling bifurcations, correct misconceptions in existing studies, and promote a deeper understanding of the complex behaviors in nonlinear dynamical systems.