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Complicated boundaries of the attraction basin in a class of three-dimensional polynomial systems. (English) Zbl 07847252


MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
34C28 Complex behavior and chaotic systems of ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI

References:

[1] Abraham, R. H., Gardini, L. & Mira, C. [1997] Chaos in Discrete Dynamical Systems: A Visual Introduction in 2 Dimensions (Springer-Verlag, NY). · Zbl 0883.58019
[2] Agliari, A., Gardini, L. & Mira, C. [2003] “ On the fractal structure of basin boundaries in two-dimensional noninvertible maps,” Int. J. Bifurcation and Chaos13, 1767-1785. · Zbl 1056.37058
[3] Aguirre, J., Viana, R. L. & Sanjuán, M. A. [2009] “ Fractal structures in nonlinear dynamics,” Rev. Mod. Phys.81, 333.
[4] Argyris, J. H., Faust, G., Haase, M. & Friedrich, R. [2015] An Exploration of Dynamical Systems and Chaos, 2nd edition (Springer, Berlin, Heidelberg). · Zbl 1314.37003
[5] Birkhoff, G. & Rota, G.-C. [1989] Ordinary Differential Equations, 4th edition (Wiley, NY). · Zbl 0183.35601
[6] Creaser, J. L., Krauskopf, B. & Osinga, H. M. [2017] “ Finding first foliation tangencies in the Lorenz system,” SIAM J. Appl. Dyn. Syst.16, 2127-2164. · Zbl 1381.37097
[7] Daza, A., Wagemakers, A., Georgeot, B., Guéry-Odelin, D. & Sanjuán, M. A. [2016] “ Basin entropy: A new tool to analyze uncertainty in dynamical systems,” Scient. Rep.6, 1-10. · Zbl 1386.37080
[8] Dong, Y., Wang, G., Iu, H. H.-C., Chen, G. & Chen, L. [2020] “ Coexisting hidden and self-excited attractors in a locally active memristor-based circuit,” Chaos30, 103123. · Zbl 1466.94059
[9] Dumortier, F., Llibre, J. & Artés, J. C. [2006] Qualitative Theory of Planar Differential Systems (Springer-Verlag, Berlin, Heidelberg). · Zbl 1110.34002
[10] Falconer, K. [2004] Fractal Geometry: Mathematical Foundations and Applications (Wiley, NY). · Zbl 1060.28005
[11] Fenichel, N. [1979] “ Geometric singular perturbation theory for ordinary differential equations,” J. Diff. Eqs.31, 53-98. · Zbl 0476.34034
[12] Feudel, U. [2008] “ Complex dynamics in multistable systems,” Int. J. Bifurcation and Chaos18, 1607-1626.
[13] Gingold, H. & Solomon, D. [2011] “ The Lorenz system has a global repeller at infinity,” J. Nonlin. Math. Phys.18, 183-189. · Zbl 1221.37033
[14] Grebogi, C., McDonald, S. W., Ott, E. & Yorke, J. A. [1983] “ Final state sensitivity: An obstruction to predictability,” Phys. Lett. A99, 415-418.
[15] Harne, R. L. & Wang, K.-W. [2017] Harnessing Bistable Structural Dynamics: For Vibration Control, Energy Harvesting and Sensing (Wiley, NY). · Zbl 1358.74002
[16] Huisman, J. & Weissing, F. J. [2001] “ Fundamental unpredictability in multispecies competition,” The Amer. Natur.157, 488-494.
[17] Katunin, A. [2017] A Concise Introduction to Hypercomplex Fractals (CRC Press, Boca Raton). · Zbl 1377.28009
[18] Khalil, H. K. & Grizzle, J. W. [2002] Nonlinear Systems, Vol. 3 (Prentice Hall, NJ). · Zbl 1003.34002
[19] Liu, J., Crawford, J. & Viola, R. [1995] “ Chaos, coexistence of attractors and fractal basin boundaries of attraction in a model system coupling activation and inhibition in parallel,” Dyn. Stab. Syst.10, 111-124. · Zbl 0840.34060
[20] Lorenz, E. N. [1963] “ Deterministic nonperiodic flow,” J. Atmosph. Sci.20, 130-141. · Zbl 1417.37129
[21] McDonald, S. W., Grebogi, C., Ott, E. & Yorke, J. A. [1985] “ Fractal basin boundaries,” Physica D17, 125-153. · Zbl 0588.58033
[22] Munkres, J. [2014] Topology, 2nd edition (Pearson, Edinburgh Gate). · Zbl 0951.54001
[23] Murdock, J. [2003] Normal Forms and Unfoldings for Local Dynamical Systems (Springer Science & Business Media). · Zbl 1014.37001
[24] Nusse, H. E. & Yorke, J. A. [2000] “ Fractal basin boundaries generated by basin cells and the geometry of mixing chaotic flows,” Phys. Rev. Lett.84, 626.
[25] Puy, A., Daza, A., Wagemakers, A. & Sanjuán, M. A. [2021] “ A test for fractal boundaries based on the basin entropy,” Commun. Nonlin. Sci. Numer. Simul.95, 105588. · Zbl 1457.37034
[26] Sparrow, C. [1982] The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, , Vol. 41 (Springer-Verlag, NY). · Zbl 0504.58001
[27] Strogatz, S. H. [2018] Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press, Boca Raton).
[28] Tucker, W. [1999] “ The Lorenz attractor exists,” Comptes Rendus de l’Académie des Sciences-Series I-Mathematics328, 1197-1202. · Zbl 0935.34050
[29] Viswanath, D. [2003] “ Symbolic dynamics and periodic orbits of the Lorenz attractor,” Nonlinearity16, 1035-1056. · Zbl 1030.37010
[30] Wiggins, S. [2003] Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer Science & Business Media). · Zbl 1027.37002
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