It is well known that the stock market, viewed as a complex, open, and nonlinear dynamical system, is affected simultaneously by many factors, such as international environment, government policies, political situation, economic situation, the public psychology over some events, some rumors, and so on, which intrinsically influence each other and make the relationships very complicated. Some of these have long influences on the market, while others have short influences on it. The arguments of synergetics, cooperation and competition among the state variables led to the case in which the system is governed by only a few slow variables. But we have no way to exactly know which and how the states govern the evolution of the system. All that we have available is the observable generated by the states, time series (index price series) from the system, which carries the information on the system of interest. How can we understand the dynamics of the system from the observable, say, the evolution of the system? We reconstruct the attractor of stock market from its stock index series with respect to delay embedding theorem (F. Takens, 1981). The attractor can then be fully unfolded in our reconstructed phase space without trajectory intersections, getting a diffeomorphic copy of the original attractor. It is suffice to say that the evolution in reconstructed phase space faithfully images, on the whole, the evolution in the original phase space, consequently laying a theoretical foundation for predicting stock index series.
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