A special type of explosion of basin boundary. (English) Zbl 1480.37040
Summary: We study a special type of explosion of a basin boundary set in an archetypal oscillator. A typical feature is that the basin boundaries change the number of basins separating at the same time. Before the explosion, a basin boundary contains some Wada points of ten basins. After the explosion, the basin boundary contains some Wada points of eighteen basins. The underlying mechanism for the explosion is investigated by the heteroclinic tangency and Lambda lemma. Basin entropy and boundary basin entropy are also used to describe the nature of basins of attraction and the basin boundary explosion.
MSC:
| 37C75 | Stability theory for smooth dynamical systems |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37G10 | Bifurcations of singular points in dynamical systems |
| 37G35 | Dynamical aspects of attractors and their bifurcations |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
References:
| [1] | Lu, G.; Smidtaite, R.; Navickas, Z.; Ragulskis, M., The effect of explosive divergence in a coupled map lattice of matrices, Chaos Solitons Fractals, 113, 308-313 (2018) · Zbl 1404.39015 |
| [2] | Scheffer, M.; Carpenter, S.; Foley, J. A.; Folke, C.; Walker, B., Catastrophic shifts in ecosystems, Nature, 413, 591-596 (2001) |
| [3] | Alley, R. B.; Marotzke, J.; Nordhaus, W. D.; Overpeck, J. T.; Peteet, D. M.; Pielke, R. A.; Pierrehumbert, R. T.; Rhines, P. B.; Stocker, T. F.; Talley, L. D.; Wallace, J. M., Abrupt climate change, Science, 299, 2005-2010 (2003) |
| [4] | Scheffer, M.; Bascompte, J.; Brock, W. A.; Brovkin, V.; Carpenter, S. R.; Dakos, V.; Held, H.; van Nes, E. H.; Rietkerk, M.; Sugihara, G., Early-warning signals for critical transitions, Nature, 461, 53-59 (2009) |
| [5] | You, Y.; Pereira, T.; Small, M.; Liu, Z.; Kurths, J., Basin of attraction determines hysteresis in explosive synchronization, Phys. Rev. Lett., 112, Article 114102 pp. (2014) |
| [6] | Karolyi, G.; Pentek, A.; Scheuring, I.; Tel, T.; Torocakai, Z., Chaotic flow: the physics of species coexistence, Proc. Natl. Acad. Sci. USA, 97, Article 13661 pp. (2000) · Zbl 0956.92032 |
| [7] | Mench, P. J.; Heitzig, J.; Marwan, N.; Kurths, J., How basin stability complements the linear-stability paradigm, Nat. Phys., 9, 89-92 (2013) |
| [8] | Leng, S.; Lin, W.; Kurths, J., Basin stability in delayed dynamics, Sci. Rep., 6, Article 21449 pp. (2016) |
| [9] | Pisarchik, A. N.; Feudel, U., Control of multistability, Phys. Rep., 540, 167-218 (2014) · Zbl 1357.34105 |
| [10] | Grebogi, C.; Ott, E.; Yorke, J. A., Basin boundary metamorphoses: changes in accessible boundary orbits, Physica D, 24, 243-262 (1987) · Zbl 0613.58018 |
| [11] | Alligood, K. T.; Lali, L. T.L.; Yorke, J. A., Metamorphoses: sudden jumps in basin boundaries, Commun. Math. Phys., 141, 1-8 (1991) · Zbl 0746.58053 |
| [12] | Nusse, H. E.; Ott, E.; Yorke, J. A., Saddle-node bifurcations on fractal basin boundaries, Phys. Rev. Lett., 75, 2482-2485 (1995) |
| [13] | Aguirre, J.; Sanjuán, M. A.F., Unpredictable behavior in the Duffing oscillator: Wada basins, Physica D, 171, 41-51 (2002) · Zbl 1008.37011 |
| [14] | Breban, R.; Nusse, H. E., On the creation of Wada basins in interval maps through fixed point tangent bifurcation, Physica D, 207, 52-63 (2005) · Zbl 1093.37012 |
| [15] | Zhang, Y.; Luo, G., Wada bifurcations and partially Wada basin boundaries in a two-dimensional cubic map, Phys. Lett. A, 377, 1274-1281 (2013) · Zbl 1290.37027 |
| [16] | Robert, C.; Alligood, K. T.; Ott, E.; Yorke, J. A., Explosions of chaotic sets, Physica D, 144, 44-61 (2000) · Zbl 0959.37042 |
| [17] | Thompson, J. M.T.; Hunt, G. W., A General Theory of Elastic Stability (1973), Wiley: Wiley London · Zbl 0351.73066 |
| [18] | Pippard, A. B., The elastic arch and its models of instability, Eur. J. Phys., 11, 359-365 (1990) |
| [19] | Cao, Q.; Wiercigroch, M.; Pavlovskaia, E. E.; Grebogi, C.; Thompson, J. M.T., Archetypal oscillator for smooth and discontinuous dynamics, Phys. Rev. E, 74, Article 046218 pp. (2006) |
| [20] | Aguirre, J.; Viana, R. L.; Sanjuán, M. A.F., Fractal structures in nonlinear dynamics, Rev. Mod. Phys., 81, 333-386 (2009) |
| [21] | Nusse, H. E.; Yorke, J. A., Wada basin boundaries and basin cells, Physica D, 90, 242-261 (1996) · Zbl 0886.58072 |
| [22] | Zhang, Y.; Lu, L. F., Basin boundaries with nested structure in a shallow arch oscillator, Nonlinear Dyn., 77, 1121-1132 (2014) |
| [23] | Daza, A.; Wagemakers, A.; Sanjuán, M. A.F., Ascertaining when a basin is Wada: the merging method, Sci. Rep., 8, 9954 (2018) |
| [24] | Kennedy, J.; Yorke, J. A., Basin of Wada, Physica D, 51, 213-225 (1991) · Zbl 0746.58054 |
| [25] | Alligood, K.; Sander, E.; Yorke, J. A., Explosions: global bifurcations at heteroclinic tangencies, Ergod. Theory Dyn. Syst., 22, 953 (2002) · Zbl 1018.37031 |
| [26] | Daza, A.; Wagemakers, A.; Georgeot, B.; Guéry-odelin, D.; Sanjuán, M. A.F., Basin entropy: a new tool to analyze uncertainty in dynamical systems, Sci. Rep., 6, Article 31416 pp. (2016) |
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