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Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures

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DOI: 10.4236/am.2011.24055    5,333 Downloads   8,832 Views   Citations


Solutions of most nonlinear differential equations have to be obtained numerically. The time series obtained by numerical integration will be a solution of the differential equation only if it is independent of the integration step h. A numerical solution is considered to have converged, when the difference between the time series for steps h and h/2 becomes smaller as h decreases. Unfortunately, this convergence criterium is unsuitable in the case of a chaotic solution, due to the extreme sensitivity to initial conditions that is characteristic of this kind of solution. We present here a criterium of convergence that involves the comparison of the attractors associated to the time series for integration time steps h and h/2. We show that the probability that the chaotic attractors associated to these time series are the same increases monotonically as the integration step h is decreased. The comparison of attractors is made using 1) the method of correlation integral, and 2) the method of statistical distance of probability distributions.

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The authors declare no conflicts of interest.

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J. de Figueiredo, L. Diambra and C. Malta, "Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures," Applied Mathematics, Vol. 2 No. 4, 2011, pp. 436-443. doi: 10.4236/am.2011.24055.


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