A geometric analysis of nonlinear dynamics and its application to financial time series

TL;DR

This paper introduces a geometric approach to analyze nonlinear oscillations, transforming differential equations into vector fields in three-dimensional space, and applies it to Japanese stock market data to identify dynamic signals.

Contribution

The paper presents a novel geometric method for analyzing nonlinear oscillations without specifying the model function, using vector fields to understand complex dynamics.

Findings

Vector fields help identify signals in stock price data.

The method captures nonlinear dynamics in financial time series.

Application demonstrates the approach's effectiveness in finance.

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 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.