  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 the system of first order ordinary differential
equations, the trajectory is embedded in $R^3$ as a curve, and thereby the time
evolution of the original state can be translated into the behavior of the
curve in $R^3$, 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 those
associated quantities can be estimated by a nonparametric filtering method. We
apply the proposed analysis to the time series of the Japanese stock price
index. The application shows that the vector field and its derivative will be
used as the tools of picking up various signals that help understanding of the
dynamic properties of the stock price index.
