Abstract
Basins of attraction take its name from hydrology, and in dynamical systems they refer to the set of initial conditions that lead to a particular final state. When different final states are possible, the predictability of the system depends on the structure of these basins. We introduce the concept of basin entropy, that aims to quantify the final state unpredictability associated to the basins. Using several paradigmatic examples from nonlinear dynamics, we dissect the meaning of this new quantity and suggest some useful applications such as the basin entropy parameter set. Then, we explain how it is possible to apply this concept to experiments with cold atoms. Previous works pointed out that chaotic dynamics could be at the heart of some interesting regimes found in the scattering of cold atoms. Here, we detail how one of the hallmarks of chaos, the appearance of fractal structures in phase space, can be detected directly from experimental measurements thanks to the basin entropy.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In this work we have normalized the region of the phase space, so that the values of the scaling box size \(\varepsilon \) in the plots are the inverse of the number of pixels used as a grid.
References
Nusse, H.E., Yorke, J.A.: Science 271, 1376 (1996)
Aguirre, J., Viana, R.L., Sanjuán, M.A.F.: Rev. Mod. Phys. 81, 333 (2009)
Kolmogorov, A.N.: Doklady Russ. Acad. Sci. 119, 861 (1959)
Sinai, Y.G.: Doklady Russ. Acad. Sci. 124, 754 (1959)
Adler, R.L., Konheim, A.G., McAndrew, M.H.: Trans. Am. Math. Soc. 114, 309 (1965)
Hunt, B.R., Ott, E.: Chaos 25, 097618 (2015)
Grebogi, C., McDonald, S.W., Ott, E., Yorke, J.A.: Phys. Lett. A 99, 415 (1983)
Grebogi, C., Kostelich, E., Ott, E., Yorke, J.A.: Phys. Lett. A 118, 448 (1986)
Menck, P.J., Heitzig, J., Marwan, N., Kurths, J.: Nat. Phys. 9, 89 (2013)
Alexander, J., Yorke, J.A., You, Z., Kan, I.: Int. J. Bifurcat. Chaos 02, 795 (1992)
Ott, E., Sommerer, J.C., Alexander, J.C., Kan, I., Yorke, J.A.: Phys. Rev. Lett. 71, 4134 (1993)
Lai, Y.-C., Winslow, R.L.: Phys. Rev. Lett. 74, 5208 (1995)
Daza, A., Wagemakers, A., Georgeot, B., Guéry-Odelin, D., Sanjuán, M.A.F.: Sci. Rep. 6, 31416 (2016)
Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems. Springer, New York (1996)
Kennedy, J., Yorke, J.A.: Phys. D 51, 213 (1991)
Daza, A., Wagemakers, A., Sanjuán, M.A.F., Yorke, J.A.: Sci. Rep. 5, 16579 (2015)
Aguirre, J., Vallejo, J.C., Sanjuán, M.A.F.: Phys. Rev. E 64, 066208 (2001)
Hnon, M., Heiles, C.: Astron. J. 69, 73 (1964)
Blesa, F., Seoane, J.M., Barrio, R., Sanjun, M.A.F.: Int. J. Bifurcat. Chaos 22, 1230010 (2012)
Epureanu, B., Greenside, H.: SIAM Rev. 40, 102 (1998)
Sanjun, M.A.F.: Phys. Rev. E 58, 4377 (1998)
Cassettari, D., Hessmo, B., Folman, R., Maier, T., Schmiedmayer, J.: Phys. Rev. Lett. 85, 5483 (2000)
Renn, M.J., Montgomery, D., Vdovin, O., Anderson, D.Z., Wieman, C.E., Cornell, E.A.: Phys. Rev. Lett. 75, 3253 (1995)
Müller, D., Cornell, E.A., Prevedelli, M., Schwindt, P.D.D., Wang, Y.-J., Anderson, D.Z.: Phys. Rev. A 63, 041602 (2001)
Houde, O., Kadio, D., Pruvost, L.: Phys. Rev. Lett. 85, 5543 (2000)
Dumke, R., Volk, M., Müther, T., Buchkremer, F.B.J., Birkl, G., Ertmer, W.: Phys. Rev. Lett. 89, 097903 (2002)
Gattobigio, G.L., Couvert, A., Georgeot, B., Guéry-Odelin, D.: Phys. Rev. Lett. 107, 254104 (2011)
Gattobigio, G.L., Couvert, A., Reinaudi, G., Georgeot, B., Guéry-Odelin, D.: Phys. Rev. Lett. 109, 030403 (2012)
Torrontegui, E., Echanobe, J., Ruschhaupt, A., Guéry-Odelin, D., Muga, J.G.: Phys. Rev. A 82, 043420 (2010)
Daza, A., Georgeot, B., Guéry-Odelin, D., Wagemakers, A.A., Sanjuán, M.A.F.: Phys. Rev. A 95, 013629 (2017)
Aguirre, J., Sanjuán, M.A.F.: Phys. Rev. E 67, 056201 (2003)
Lai, Y.C., Tél, T.: Transient Chaos. Springer, New York (2011)
Seoane, J.M., Sanjuán, M.A.F., Lai, Y.C.: Phys. Rev. E 76, 016208 (2007)
Seoane, J.M., Huang, L., Sanjuán, M.A.F., Lai, Y.C.: Phys. Rev. E 79, 047202 (2009)
Motter, A.E., Lai, Y.C.: Phys. Rev. E 65, 015205 (2001)
Seaman, B.T., Krämer, M., Anderson, D.Z., Holland, M.J.: Phys. Rev. A 75, 023615 (2007); Ryu, C., Boshier, M.G.: New J. Phys. 17, 092002 (2015); Caliga, S.C., Straatsma, C.J.E., Anderson, D.Z.: New J. Phys. 18, 025010 (2016)
Acknowledgements
This work was supported by the Spanish Ministry of Economy and Competitiveness under Project No. FIS2013-40653-P and by the Spanish State Research Agency (AEI) and the European Regional Development Fund (FEDER) under Project No. FIS2016-76883-P. MAFS acknowledges the jointly sponsored financial support by the Fulbright Program and the Spanish Ministry of Education (Program No. FMECD-ST-2016). Financial support from the Programme Investissements d’Avenir under the program ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT is also acknowledged.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Daza, A., Wagemakers, A., Georgeot, B., Guéry-Odelin, D., Sanjuán, M.A.F. (2018). Basin Entropy, a Measure of Final State Unpredictability and Its Application to the Chaotic Scattering of Cold Atoms. In: Edelman, M., Macau, E., Sanjuan, M. (eds) Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-68109-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-68109-2_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68108-5
Online ISBN: 978-3-319-68109-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)