A precise calculation of bifurcation points for periodic solution in nonlinear dynamical systems

• This paper presented a precise yet efficient algorithm to determine bifurcation points of periodic solutions in nonlinear dynamical systems.
• Period doubling and symmetry breaking points were obtained to an accuracy of 10−11.
• The bifurcation points can be directly determined without a tedious trail and error process.

Calculations and analysis on bifurcations of periodic solutions has played a pivotal role in nonlinear dynamics researches. This paper presents an efficient and precise approach, based on the incremental harmonic balance (IHB) method, to exactly determine the critical points at which periodic solutions exhibit bifurcations. The presented method is based on a fact that, some zero harmonic coefficients vary as non-zeros during the bifurcation of a periodic solution. Equating one of these coefficients to an infinitesimal quantity, the bifurcation value can be determined by the IHB method with the control parameter varying. Numerical examples show that, the critical values for the control parameter where a symmetry breaking or period doubling happens can be calculated very accurately. Importantly, each bifurcation value can be directly obtained without a tedious trail and error process.