Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures

References

[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