Threshold Autoregression, Limit Cycles and Cyclical Data

TLDR

The notion of a limit cycle, a phenomenon exclusive to nonlinear systems, is central to modeling cyclical data. The authors aim to show that threshold autoregressive models can capture limit cycles and provide a discrete‑time definition. They demonstrate this by proving the models’ generality, interpreting the threshold value, and presenting simulations and real‑data analyses that reveal characteristic nonlinear vibration features.

Abstract

Summary The notion of a limit cycle, which can only exist in a non-linear system, plays the key role in the modelling of cyclical data. We have shown that the class of threshold autoregressive models is general enough to capture this notion, a definition of which in discrete time is proposed. The threshold value has an interesting interpretation. Simulation results are presented which demonstrate that this new class of models exhibits some well-known features of non-linear vibrations. Detailed analyses of several real data sets are discussed.