"Periodic Sequences" Embedded in Aperiodic Motions in the Lorenz System

[摘要] The concept of "periodic sequences" in aperiodic motions is demonstrated in order to describe the global structure of the strange attractor by using the Lorenz system of three variables. We define each periodic sequence as a time interval when the trajectory passes in the vicinity of a periodic solution (PS) or a "pseudo-periodic solution (PPS)" with a relatively short period ; the PPS, which is characterized by almost periodic motions, bifurcates from a limit point of a PS. Hereafter, we use the abbreviation "MPP (local minimum point based on the periodicity)" to refer to either a PS or a PPS. For a wide range of bifurcation parameter values, the statistical significance of the relation between each periodic sequence and an MPP is obtained. Some MPPs are preferentially selected to generate periodic sequences in aperiodic motions. This selection does not depend on the linear stability of each MPP. The probability that each periodic sequence persists over n cycles is well expressed by exp(-n/τ), where τ is a characteristic time. For periodic sequences near each PS, the characteristic time is also determined primarily by the linear stability of the PS. These facts support the relation between each periodic sequence and an MPP. Thus, the aperiodic motion can be grasped by the transition between MPPs embedded in the strange attractor.

[学科分类] 大气科学