Frequency-domain computation of Lyapunov-Perron transformation with detection of quasi-periodic solution bifurcation

Abstract

Lyapunov-Perron (LP) transformation is well-known as an indispensible technique to reduce a linear quasi-periodic (QP) system, under certain conditions, into a time-invariant one, which we call the reduced system. The reducibility of QP system received a lot of qualitative studies, however quite few research focused on the quantification of the LP transformation itself. The major obstacle probably consists in the calculation of a QP matrix describing the transformation. In this study we will propose a semi-analytical approach, based on frequency-domain analysis, to determine the LP transformation precisely and efficiently. Our approach is divided into two steps. Firstly a constant matrix is determined to represent the reduced system, utilizing the circularly distributed multipliers of the QP system provided by a frequency-domain approach established recently by the authors. With the constant matrix, a linear and solvable QP system governing the transformation matrix is deduced and solved via harmonic balancing, in which the harmonic coefficients are provided by linear algebraic equations. Numerical examples show the semi-analytical results are very accurate for both the constant and the QP matrix, representing the reduced system and the LP transformation, respectively.

As an application, the proposed approach provides us with a convenient way to detect the bifurcation types of QP responses. For instance, the harmonic components of the QP matrix are analyzed for distinguishing saddle-node and symmetry-breaking bifurcations. For a special example where the QP system cannot be reduced via the routine LP transformation, interestingly, it turns out the system reduction can still be realized via an extended transformation. The harmonic components of the QP matrix in the extended transformation can also identify period-doubling bifurcation. Lastly the proposed approach is shown to be capable of detecting QP Hopf bifurcation, exemplified by a van der Pol-Rossler coupled system.