×

zbMATH — the first resource for mathematics

Classifying and quantifying basins of attraction. (English) Zbl 1374.37026
Summary: A scheme is proposed to classify the basins for attractors of dynamical systems in arbitrary dimensions. There are four basic classes depending on their size and extent, and each class can be further quantified to facilitate comparisons. The calculation uses a Monte Carlo method and is applied to numerous common dissipative chaotic maps and flows in various dimensions.
©2015 American Institute of Physics

MSC:
37B25 Stability of topological dynamical systems
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
Software:
Dynamics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Leonov, G. A.; Kuznetsov, N. V.; Vagaitsev, V. I., Phys. Lett. A, 375, 2230, (2011) · Zbl 1242.34102
[2] Leonov, G. A.; Kuznetsov, N. V., Int. J. Bifurcat. Chaos, 23, 1330002, (2013)
[3] McDonald, S. W.; Grebogi, C.; Ott, E.; Yorke, J. A., Physica D, 17, 125, (1985) · Zbl 0588.58033
[4] Mira, C.; Fournier-Prunaret, D.; Gardini, L.; Kawakami, H.; Cathala, J. C., Int. J. Bifurcat. Chaos, 4, 343, (1994) · Zbl 0818.58032
[5] Ott, E.; Sommerer, J. C.; Alexander, J. C.; Kan, I.; Yorke, J. A., Phys. Rev. Lett., 71, 4134, (1993) · Zbl 0972.37514
[6] Metropolis, N.; Ulam, S., J. Am. Stat. Assoc., 44, 335, (1949)
[7] May, R., Nature, 261, 459, (1976) · Zbl 1369.37088
[8] Grassberger, P., J. Stat. Phys., 26, 173, (1981)
[9] Molaie, M.; Jafari, S.; Sprott, J. C.; Golpayegani, S. M. R. H., Int. J. Bifurcat. Chaos, 23, 1350188, (2013) · Zbl 1284.34064
[10] Sprott, J. C., Phys. Lett. A, 378, 1361, (2014) · Zbl 1323.37022
[11] Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A., Phys. Nonlinear Phenom., 16, 285, (1985) · Zbl 0585.58037
[12] Ikeda, K., Opt. Commun., 30, 257, (1979)
[13] Henon, M., Comm. Math. Phys., 50, 69, (1976) · Zbl 0576.58018
[14] Nusse, H. E.; Yorke, J. A., Dynamics: Numerical Explorations, (1994), Springer: Springer, New York · Zbl 0805.58007
[15] Devaney, R. L., A First Course in Chaotic Dynamical Systems: Theory and Experiment, (1992), Addison-Wesley-Longman: Addison-Wesley-Longman, Reading, MA · Zbl 0768.58001
[16] Mandelbrot, B. B., Ann. N. Y. Acad. Sci., 357, 249, (1980)
[17] Lozi, R., J. Phys. (Paris), 39, C5-9, (1978)
[18] Sprott, J. C., Chaos and Time-Series Analysis, (2003), Oxford Unversity Press: Oxford Unversity Press, Oxford
[19] Lorenz, E. N., J. Atmos. Sci., 20, 130, (1963) · Zbl 1417.37129
[20] Chen, G.; Ueda, T., Int. J. Bifurcat. Chaos, 9, 1465, (1999) · Zbl 0962.37013
[21] Strogatz, S. H., Nonlinear Dynamics and Chaos, (2015), Westview Press: Westview Press, Philadelphia, PA
[22] Rössler, O. E., Phys. Lett. A, 57, 397, (1976) · Zbl 1371.37062
[23] Sprott, J. C., Phys. Lett. A, 228, 271, (1997) · Zbl 1043.37504
[24] Rössler, O. E., Phys. Lett. A, 71, 155, (1979) · Zbl 0996.37502
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.