Parameter uncertainties in models of equivariant dynamical systems. (English) Zbl 0890.58037

Summary: We examine the advantages of using equivariant versus nonequivariant phenomenological models for chaotic systems that possess an inversion symmetry. Numerical experiments used the minimum description length (MDL) criteria for model selection. They indicated that the selection between equivariant versus nonequivariant models should be made on an a priori basis. We also examine the relationship between the optimal truncation accuracy of the parameters of the model (as predicted by the MDL criteria) and the true uncertainty in the parameters of the model. Numerical experiments indicate that the true uncertainties are poorly approximated by the optimal truncation accuracy. They also indicate that the true uncertainties may be useful for determining the presence of an inversion symmetry in reconstructed attractors.


37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37C80 Symmetries, equivariant dynamical systems (MSC2010)
37N99 Applications of dynamical systems
Full Text: DOI


[1] Geona, M.; Lentini, F.; Cimagalli, V., Phys. Rev. A, 44, 3496 (1991)
[2] (Stochastic Complexity in Statistical Inquiry (1989), World Scientific: World Scientific NJ) · Zbl 0800.68508
[3] Letellier, C.; Le Sceller, L.; Maréchal, E.; Dutertre, P.; Maheu, B.; Gouesbet, G.; Fei, Z.; Hudson, J. L., Phys. Rev. E, 51, 4262 (1995)
[4] Buchler, J. R.; Serre, T.; Kolláth, Z.; Mattei, J., Phys. Rev. Lett., 74, 842 (1995)
[5] Judd, K.; Mees, A., Physica D, 82, 426 (1995)
[6] Brown, R.; Rulkov, N. F.; Tracy, E. R., Phys. Rev. E, 49, 3784 (1994)
[7] Brown, R.; Rulkov, N. F.; Tufillaro, N. B., Phys. Rev. E, 50, 4488 (1994)
[8] Barany, E.; Delnitz, M.; Golubitsky, M., Physica D, 67, 66 (1993)
[9] King, G. P.; Stewart, I., (Ames, W. F.; Rogers, C., Nonlinear Equations in the Applied Sciences (1992), Academic Press: Academic Press New York)
[10] Letellier, C.; Gouesbet, G., Phys. Rev. E, 52, 4754 (1995)
[11] Kitano, M.; Yabuzaki, T.; Ogawa, T., Phys. Rev. A, 29, 1288 (1984)
[12] Abarbanel, H. D.I.; Brown, R.; Sidorowich, J. J.; Tsimring, L. S., Rev. Mod. Phys., 65, 1331 (1993)
[13] Brown, R., INLS Report (1 April 1993), (unpublished)
[14] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes (1992), Cambridge University Press: Cambridge University Press New York · Zbl 0778.65003
[15] C. Letellier, Private communication.; C. Letellier, Private communication.
[16] Grebogi, C.; Ott, E.; Yorke, J. A., Physica D, 7, 181 (1983)
[17] Rulkov, N. F.; Volkovskii, A. R., (Proc. Exploiting Chaos and Nonlinearities, Vol. 2038 (1993), SPIE Press)
[18] Ditto, W. L.; Rauseo, S. N.; Heagy, J.; Ott, E., Phys. Rev. Lett., 65, 533 (1990)
[19] Kennel, M.; Brown, R.; Abarbanel, H. D.I., Phys. Rev. A, 45, 3403 (1992)
[20] Gouesbet, G., Phys. Rev. A, 43, 5321 (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.