Efforts to simulate turbulence in the financial markets include experiments with the logistic equation: x(t)=kappa x(t-1)[1-x(t-1)], with 0 < x(t)<1 and 0 = < kappa < 4. Visual investigation of the logistic equation show the various stability and instability regimes for the various value of the Feigenbaum number kappa. Visualizations for t=20 observations provide clear demonstrations of the stability regimes. We visually investigate these regimes in more detail in the t=101-110 range. For 0 < kappa < 3, the process settles to a unique stable equilibrium. For 3= < kappa < 3.6 the process bifurcates, or, as colored visualization shows but not black-and-white, its pitchfork bifurcation branches 'bang-bang' switch between two regimes. For 3.6= < kappa = < 4.0 the process becomes chaotic, i.e., deterministically random. In this regime are windows of stability, e.g., at kappa=3+2sqrt=3.8284. At kappa=4, pure chaos, the process is extremely sensitive to initial values, as visually is clearly demonstrated. We increase the number of observations to t=1000 and compute the homogeneous Hurst exponent of the process at kappa=4: H=0.004, indicating that x(t) is blue noise, i.e., extreme anti-persistent. A histogram shows a highly platykurtic distribution of x(t), with an imploded 'mode,' with extremely fat tails higher than the 'mode,' against the reflecting values at x=0 and x=1. Several plots of the state directory of the system in the (x(t),x(t-1))- space trace out the parabolic strange attractor. Although the strange attractor is a well-defined parabole, the points on the attarctor set are deterministically random and unpredictable.
Download Info
To download:
If you experience problems downloading a file, check if you have the
proper application to
view it first. Information about this may be contained
in the File-Format links below. In case of further problems read
the IDEAS help
page. Note that these files are not on the IDEAS
site. Please be patient as the files may be large.
Publisher Info
Paper provided by EconWPA in its series Finance with number
0409035.
Find related papers by JEL classification: C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Statistical Simulation Methods C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Other C33 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Models with Panel Data C49 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Other
This paper has been announced in the following NEP Reports:
References listed on IDEAS Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
Cited by: (explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)