Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures

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.

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.

Cite this paper

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.

nullJ. de Figueiredo, L. Diambra and C. Malta, "Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures,"

References

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[6] 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

[7] 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

[8] 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.

[9] 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

[10] A. Kolmogorov, “Sulla Determinazione Empirica di um Legge di Distribuizione,” Giornale Dell'Instituto Italiano Degli Attuari, Vol. 4, 1933, pp. 83-91.

[11] 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.

[12] 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

[13] 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

[14] I. Csiszár, “Statistical Decision Functions and Random Processes,” Proceedings of 7th Prague Conference on Information Theory, Prague, 1974, pp. 73-86.

[15] 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

[16] 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

[1] 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

[2] 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.

[3] 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.

[4] 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

[5] 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

[6] 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

[7] 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

[8] 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.

[9] 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

[10] A. Kolmogorov, “Sulla Determinazione Empirica di um Legge di Distribuizione,” Giornale Dell'Instituto Italiano Degli Attuari, Vol. 4, 1933, pp. 83-91.

[11] 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.

[12] 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

[13] 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

[14] I. Csiszár, “Statistical Decision Functions and Random Processes,” Proceedings of 7th Prague Conference on Information Theory, Prague, 1974, pp. 73-86.

[15] 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

[16] 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