ABSTRACT 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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nullJ. 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.
 J. Teixeira, C. A. Reynolds and K. Judd, “Time Step Sensitivity of Nonlinear Atmospheric Models: Numerical Convergence, Truncation Error Growth, and Ensemble Design,” Journal of the Atmospheric Sciences, Vol. 64, No. 1, January 2007, pp. 175-189. doi:10.1175/JAS3824.1
 L. S. Yao and D. Hughes, “Comment on ‘Time Step Sensitivity of Nonlinear Atmospheric Models: Numerical Convergence, Truncation Error Growth, and Ensemble Design’,” Journal of the Atmospheric Sciences, Vol. 65, No. 2, February 2007, pp. 681-682.
 L. S. Yao and D. Hughes, “Comment on ‘Computational Periodicity as Observed in a Simple System, by E. N. Lorenz’,” Tellus, Vol. 60, No. 4, August 2008, pp. 803- 805.
 T. D. Sauer, “Shadowing Breakdown and Large Errors in Dynamical Simulations of Physical Systems,” Physical Review E, Vol. 65, No. 3, March 2002, pp. 036220-1-5. doi:10.1103/PhysRevE.65.036220
 L. Glass, A. Beuter and D. Larocque, “Time Delays, Oscillations and Chaos in Physiological Control Systems,” Mathematical Biosciences, Vol. 90, No. 1-2, July-August 1988, pp. 111-125. doi:10.1016/0025-5564(88)90060-0
 L. Glass and C. P. Malta, “Chaos in Multi-Looped Negative Feedback Systems,” Journal of Theoretical Biology, Vol. 145, No. 2, July 1990, pp. 217-223. doi:10.1016/S0022-5193(05)80127-4
 J. C. B. de Figueiredo, L. Diambra, L. Glass and C. P. Malta, “Chaos in Two-Loop Negative Feedback Systems,” Physical Review E, Vol. 65, No. 5, May 2002, pp. 051905-1-8. doi:10.1103/PhysRevE.65.051905
 C. P. Malta and M. L. S. Teles, “Nonlinear Delay Differential Equations: Comparison of Integration Methods,” International Journal of Applied Mathematics, Vol. 3, No 4, July 2000, pp. 379-395.
 P. Grassberger and I. Procaccia, “Measuring the Strangeness of Strange Attractors,” Physica D: Nonlinear Phenomena, Vol. 9, No. 1-2, October 1983, pp. 189-208. doi:10.1016/0167-2789(83)90298-1
 A. Kolmogorov, “Sulla Determinazione Empirica di um Legge di Distribuizione,” Giornale Dell'Instituto Italiano Degli Attuari, Vol. 4, 1933, pp. 83-91.
 N. Smirnov, “On the Estimation of Discrepancy between Empirical Curves of Distribution for Two Independent Samples,” Bulletin Mathématique de L′Université de Moscow, Vol. 2, No. 2, 1939, pp. 3-11.
 L. Diambra, “Divergence Measure between Chaotic Attractors,” Physical Review E, Vol. 64, No. 3, September 2001, pp. 035202-1-5. doi:10.1103/PhysRevE.64.035202
 A. M. Albano, P. E. Rapp and A. Passamante, “Kolmogorov-Smirnov Test Distinguishes Attractors with Similar Dimensions,” Physical Review E, Vol. 52, No. 1, July 1995, pp. 196-205. doi:10.1103/PhysRevE.52.196
 I. Csiszár, “Statistical Decision Functions and Random Processes,” Proceedings of 7th Prague Conference on Information Theory, Prague, 1974, pp. 73-86.
 C. Tsallis, “Possible Generalization of Boltzmann-Gibbs Statistics,” Journal of statistical physics, Vol. 52, No. 1-2, July 1988, pp. 479-487. doi:10.1007/BF01016429
 G. J. Stienstra, J. Nijenhuis, T. Kroezen, C. M. van den Bleek and J. R. van Ommen, “Monitoring Slurry-Loop Reactors for Early Detection of Hydrodynamic Instabilities,” Chemical Engineering and Processing, Vol. 44, No. 9, September 2005, pp. 959-968. doi:10.1016/j.cep.2005.01.001