Abstract
A geometric method to analyze nonlinear oscillations is discussed. We consider a nonlinear oscillation modeled by a second-order ordinary differential equation without specifying the function form. By transforming the differential equation into a system of first-order ordinary differential equations, the trajectory is embedded in R3 as a curve, and thereby the time evolution of the original state can be translated into the behavior of the curve in R3, or the vector field along the curve. We analyze the vector field to investigate the dynamic properties of a nonlinear oscillation. While the function form of the model is unspecified, the vector fields and associated quantities can be estimated by a nonparametric filtering method. Estimates of vector field and its derivative will catch signals that help understanding of the dynamic properties of a state of our interest. Applying the proposed analysis to the time series of the Japanese stock price index, the vector fields and its derivative indicate that quite inefficient behaviors in a geometric sense occur from 2008 to 2009, corresponding to the years supposedly affected by the Lehman’s collapse.
Published Version
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