×

zbMATH — the first resource for mathematics

Riddled basins of attraction in systems exhibiting extreme events. (English) Zbl 1390.34150
Summary: Using a system of two FitzHugh-Nagumo units, we demonstrate the occurrence of riddled basins of attraction in delay-coupled systems as the coupling between the units is increased. We characterize riddled basins using the uncertainty exponent which is a measure of the dimensions of the basin boundary. Additionally, we show that the phase space can be partitioned into pure and mixed regions, where initial conditions in the pure regions certainly avoid the generation of extreme events, while initial conditions in the mixed region may or may not exhibit such events. This implies that any tiny perturbation of initial conditions in the mixed region could yield the emergence of extreme events because the latter state possesses a riddled basin of attraction.
©2018 American Institute of Physics

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ansmann, G.; Karnatak, R.; Lehnertz, K.; Feudel, U., Extreme events in excitable systems and mechanisms of their generation, Phys. Rev. E, 88, 052911, (2013)
[2] Karnatak, R.; Ansmann, G.; Feudel, U.; Lehnertz, K., Route to extreme events in excitable systems, Phys. Rev. E, 90, 022917, (2014)
[3] Helbing, D., Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73, 1067-1141, (2001)
[4] Feigenbaum, J., A statistical analysis of log-periodic precursors to financial crashes, Quant. Finance, 1, 346-360, (2001) · Zbl 1405.62139
[5] Dobson, I.; Carreras, B. A.; Lynch, V. E.; Newman, D. E., Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization, Chaos, 17, 026103, (2007) · Zbl 1159.37344
[6] Bunde, A.; Kropp, J.; Schellnhuber, H., The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes, (2002), Springer: Springer, Berlin, Heidelberg
[7] Akhmediev, N.; Ankiewicz, A.; Taki, M., Waves that appear from nowhere and disappear without a trace, Phys. Lett. A, 373, 675-678, (2009) · Zbl 1227.76010
[8] Chabchoub, A.; Hoffmann, N. P.; Akhmediev, N., Rogue wave observation in a water wave tank, Phys. Rev. Lett., 106, 204502, (2011)
[9] Chabchoub, A.; Hoffmann, N.; Onorato, M.; Akhmediev, N., Super rogue waves: Observation of a higher-order breather in water waves, Phys. Rev. X, 2, 011015, (2012)
[10] Bonatto, C.; Feyereisen, M.; Barland, S.; Giudici, M.; Masoller, C.; Leite, J. R. R.; Tredicce, J. R., Deterministic optical rogue waves, Phys. Rev. Lett., 107, 053901, (2011)
[11] Akhmediev, N., Roadmap on optical rogue waves and extreme events, J. Opt., 18, 063001, (2016)
[12] Pisarchik, A. N.; Jaimes-Re√°tegui, R.; Sevilla-Escoboza, R.; Huerta-Cuellar, G.; Taki, M., Rogue waves in a multistable system, Phys. Rev. Lett., 107, 274101, (2011)
[13] Bosco, A. K. D.; Wolfersberger, D.; Sciamanna, M., Extreme events in time-delayed nonlinear optics, Opt. Lett., 38, 703-705, (2013)
[14] Bialonski, S.; Ansmann, G.; Kantz, H., Data-driven prediction and prevention of extreme events in a spatially extended excitable system, Phys. Rev. E, 92, 042910, (2015)
[15] Bialonski, S.; Caron, D. A.; Schloen, J.; Feudel, U.; Kantz, H.; Moorthi, S. D., Phytoplankton dynamics in the southern California bight indicate a complex mixture of transport and biology, J. Plankton Res., 38, 1077, (2016)
[16] Lehnertz, K., Epilepsy and nonlinear dynamics, J. Biol. Phys., 34, 253-266, (2008)
[17] Lehnertz, K.; Albeverio, S.; Jentsch, V.; Kantz, H., Epilepsy: Extreme events in the human brain, Extreme Events in Nature and Society, 123-143, (2006), Springer: Springer, Berlin, Heidelberg
[18] Kim, J.-W.; Ott, E., Statistics and characteristics of spatiotemporally rare intense events in complex Ginzburg-Landau models, Phys. Rev. E, 67, 026203, (2003)
[19] Reinoso, J. A.; Zamora-Munt, J.; Masoller, C., Extreme intensity pulses in a semiconductor laser with a short external cavity, Phys. Rev. E, 87, 062913, (2013)
[20] Zamora-Munt, J.; Garbin, B.; Barland, S.; Giudici, M.; Leite, J. R. R.; Masoller, C.; Tredicce, J. R., Rogue waves in optically injected lasers: Origin, predictability, and suppression, Phys. Rev. A, 87, 035802, (2013)
[21] Rothkegel, A.; Lehnertz, K., Irregular macroscopic dynamics due to chimera states in small-world networks of pulse-coupled oscillators, New J. Phys., 16, 055006, (2014)
[22] Saha, A.; Feudel, U., Extreme events in FitzHugh-Nagumo oscillators coupled with two time delays, Phys. Rev. E, 95, 062219, (2017)
[23] Vandermeer, J.; Stone, L.; Blasius, B., Categories of chaos and fractal basin boundaries in forced predatorprey models, Chaos, Solitons Fractals, 12, 265-276, (2001) · Zbl 0976.92033
[24] Grebogi, C.; Ott, E.; Yorke, J. A., Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics, Nonlinear Physics for Beginners, 92-146, (2012), World Scientific
[25] McDonald, S. W.; Grebogi, C.; Ott, E.; Yorke, J. A., Fractal basin boundaries, Physica D, 17, 125-153, (1985) · Zbl 0588.58033
[26] Nusse, H. E.; Yorke, J. A., Wada basin boundaries and basin cells, Physica D, 90, 242-261, (1996) · Zbl 0886.58072
[27] Poon, L.; Campos, J.; Ott, E.; Grebogi, C., Wada basin boundaries in chaotic scattering, Int. J. Bifurcation Chaos, 06, 251-265, (1996) · Zbl 0870.58069
[28] Sommerer, J. C.; Ott, E., Intermingled basins of attraction: Uncomputability in a simple physical system, Phys. Lett. A, 214, 243-251, (1996) · Zbl 0972.37550
[29] Ding, M.; Yang, W., Observation of intermingled basins in coupled oscillators exhibiting synchronized chaos, Phys. Rev. E, 54, 2489-2494, (1996)
[30] Lai, Y.-C.; Grebogi, C., Intermingled basins and two-state on-off intermittency, Phys. Rev. E, 52, R3313-R3316, (1995)
[31] Alexander, J.; Yorke, J. A.; You, Z.; Kan, I., Riddled basins, Int. J. Bifurcation Chaos, 02, 795-813, (1992) · Zbl 0870.58046
[32] Heagy, J. F.; Carroll, T. L.; Pecora, L. M., Experimental and numerical evidence for riddled basins in coupled chaotic systems, Phys. Rev. Lett., 73, 3528-3531, (1994)
[33] Ott, E.; Alexander, J.; Kan, I.; Sommerer, J.; Yorke, J., The transition to chaotic attractors with riddled basins, Physica D, 76, 384-410, (1994) · Zbl 0820.58043
[34] Grebogi, C.; Ott, E.; Yorke, J. A., Fractal basin boundaries, long-lived chaotic transients, and unstable-unstable pair bifurcation, Phys. Rev. Lett., 50, 935-938, (1983)
[35] Kennedy, J.; Yorke, J. A., Basins of Wada, Physica D, 51, 213-225, (1991) · Zbl 0746.58054
[36] Ott, E.; Sommerer, J. C., Blowout bifurcations: The occurrence of riddled basins and on-off intermittency, Phys. Lett. A, 188, 39-47, (1994)
[37] Ott, E.; Sommerer, J. C.; Alexander, J. C.; Kan, I.; Yorke, J. A., Scaling behavior of chaotic systems with riddled basins, Phys. Rev. Lett., 71, 4134-4137, (1993) · Zbl 0972.37514
[38] Ashwin, P.; Buescu, J.; Stewart, I., Bubbling of attractors and synchronisation of chaotic oscillators, Phys. Lett. A, 193, 126-139, (1994) · Zbl 0959.37508
[39] Ashwin, P.; Buescu, J.; Stewart, I., From attractor to chaotic saddle: A tale of transverse instability, Nonlinearity, 9, 703, (1996) · Zbl 0887.58034
[40] Cazelles, B., Dynamics with riddled basins of attraction in models of interacting populations, Chaos, Solitons Fractals, 12, 301-311, (2001) · Zbl 0986.92034
[41] Maistrenko, Y. L.; Maistrenko, V. L.; Popovich, A.; Mosekilde, E., Transverse instability and riddled basins in a system of two coupled logistic maps, Phys. Rev. E, 57, 2713-2724, (1998)
[42] Maistrenko, Y.; Kapitaniak, T.; Szuminski, P., Locally and globally riddled basins in two coupled piecewise-linear maps, Phys. Rev. E, 56, 6393-6399, (1997)
[43] Kapitaniak, T., Experimental observation of riddled behaviour (electronic system), J. Phys. A: Math. Gen., 28, L63, (1995) · Zbl 0851.34036
[44] Ujjwal, S. R.; Punetha, N.; Ramaswamy, R.; Agrawal, M.; Prasad, A., Driving-induced multistability in coupled chaotic oscillators: Symmetries and riddled basins, Chaos, 26, 063111, (2016) · Zbl 1374.34181
[45] Ansmann, G., Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE, (2017) · Zbl 1390.34005
[46] Shampine, L.; Thompson, S., Solving DDEs in Matlab, Appl. Numer. Math., 37, 441-458, (2001) · Zbl 0983.65079
[47] Grebogi, C.; McDonald, S. W.; Ott, E.; Yorke, J. A., Final state sensitivity: An obstruction to predictability, Phys. Lett. A, 99, 415-418, (1983)
[48] Milnor, J.; Hunt, B. R.; Li, T.-Y.; Kennedy, J. A.; Nusse, H. E., On the concept of attractor, The Theory of Chaotic Attractors, 243-264, (2004), Springer: Springer, New York, NY
[49] Milnor, J., On the concept of attractor: Correction and remarks, Commun. Math. Phys., 102, 517-519, (1985) · Zbl 0602.58030
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.