## A Random-Walk or Color-Chaos on the Stock Market?

The random-walk (white-noise) model and the harmonic model are two polar models in linear systems. A model in between is color chaos which generates irregular oscillations with a narrow frequency (color) band. Time-frequency analysis is introduced for evolutionary time series analysis.The deterministic component from noisy data can be recovered by a time-variant filter in Gabor space. The characteristic frequency is calculated from the Wigner decomposed distribution series. It is found that about seventy percent of fluctuations in Standard & Poor stock price indexes, such as the FSPCOM and FSDXP monthly series, detrended by the Hodrick-Prescott (HP) filter, can be explained by deterministic color chaos. The characteristic period of persistent cycles is around three to four years. Their correlation dimension is about 2.5.

The existence of persistent chaotic cycles reveals new perspective of market resilience and new sources of economic uncertainties. The nonlinear pattern in the stock market may not be wiped out by market competition under nonequilibrium situations with trend evolution and frequency shifts. The color-chaos model of stock-market movements may establish a link between business-cycle theory and asset-pricing theory.

A Random Walk or Color Chaos on the Stock Market?Time Frequency Analysis of S P Indexes