The three-dimensional generalized Hénon map: bifurcations and attractors. (English) Zbl 07879632
References:
| [1] | Hénon, M., A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50, 69-77, 1976 · Zbl 0576.58018 · doi:10.1007/BF01608556 |
| [2] | Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences Vol. 42 (Springer-Verlag, New York, 2002). · Zbl 0515.34001 |
| [3] | Kuznetsov, Y., Elements of Bifurcation Theory, 3rd ed., Applied Mathematical Sciences Vol. 112 (Springer-Verlag, New York, 2004). · Zbl 1082.37002 |
| [4] | Gonchenko, S.; Ovsyannikov, I. I.; Simó, C.; Turaev, D., Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Int. J. Bifurcation Chaos, 15, 3493-3508, 2005 · Zbl 1097.37023 · doi:10.1142/S0218127405014180 |
| [5] | Devaney, R.; Nitecki, Z., Shift automorphisms in the Hénon mapping, Commun. Math. Phys., 67, 137-146, 1979 · Zbl 0414.58028 · doi:10.1007/BF01221362 |
| [6] | Benedicks, M.; Carleson, L., The dynamics of the Hénon map, Ann. Math., 133, 73-169, 1991 · Zbl 0724.58042 · doi:10.2307/2944326 |
| [7] | Sterling, D.; Meiss, J., Computing periodic orbits using the anti-integrable limit, Phys. Lett. A, 241, 46-52, 1998 · Zbl 0949.37012 · doi:10.1016/S0375-9601(98)00094-2 |
| [8] | Lomelí, H.; Meiss, J., Quadratic volume-preserving maps, Nonlinearity, 11, 557-574, 1998 · Zbl 0902.58010 · doi:10.1088/0951-7715/11/3/009 |
| [9] | Elhadj, Z.; Sprott, J., Classification of three-dimensional quadratic diffeomorphisms with constant Jacobian, Front. Phys. Chin., 4, 111-121, 2009 · doi:10.1007/s11467-009-0005-y |
| [10] | Lenz, K.; Lomelí, H.; Meiss, J., Quadratic volume preserving maps: An extension of a result of Moser, Regul. Chaotic Dyn., 3, 122-130, 1999 · Zbl 0997.37028 · doi:10.1070/rd1998v003n03ABEH000085 |
| [11] | Dullin, H.; Meiss, J., Nilpotent normal forms for a divergence-free vector fields and volume-preserving maps, Physica D, 237, 155-166, 2008 · Zbl 1141.37005 · doi:10.1016/j.physd.2007.08.014 |
| [12] | Dullin, H.; Meiss, J., Quadratic volume-preserving maps: Invariant circles and bifurcations, SIAM J. Appl. Dyn. Syst., 8, 76-128, 2009 · Zbl 1183.37090 · doi:10.1137/080728160 |
| [13] | Gonchenko, S. V.; Meiss, J.; Ovsyannikov, I., Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regul. Chaotic Dyn., 11, 191-212, 2006 · Zbl 1164.37306 · doi:10.1070/RD2006v011n02ABEH000345 |
| [14] | Hampton, A.; Meiss, J., Anti-integrability for three-dimensional quadratic maps, SIAM J. Appl. Dyn. Syst., 21, 650-675, 2022 · Zbl 1528.37056 · doi:10.1137/21M1433289 |
| [15] | Turaev, D.; Shilnikov, L., An example of a wild strange attractor, Mat. Sb., 189, 137-160, 1998 · Zbl 0927.37017 · doi:10.1070/SM1998v189n02ABEH000300 |
| [16] | Gonchenko, A. S.; Gonchenko, S., Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps, Phys. D, 337, 43-57, 2016 · Zbl 1376.37047 · doi:10.1016/j.physd.2016.07.006 |
| [17] | Gonchenko, S.; Simo, C.; Vieiro, A., Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight, Nonlinearity, 26, 621-678, 2013 · Zbl 1286.37023 · doi:10.1088/0951-7715/26/3/621 |
| [18] | Gonchenko, S. V.; Ovsyannikov, I., Homoclinic tangencies to resonant saddles and discrete Lorenz attractors, Discrete Contin. Dyn. Syst. S, 10, 273-288, 2017 · Zbl 1372.37049 · doi:10.3934/dcdss.2017013 |
| [19] | Gonchenko, A. S.; Gonchenko, M. S.; Kozlov, A. D.; Samylina, E. A., On scenarios of the onset of homoclinic attractors in three-dimensional non-orientable maps, Chaos, 31, 043122, 2021 · Zbl 1468.37034 · doi:10.1063/5.0039870 |
| [20] | Zhao, M.; Li, C.; Wang, J.; Feng, Z., Bifurcation analysis of the three-dimensional Hénon map, Discrete Contin. Dyn. Syst. S, 10, 625-645, 2017 · doi:10.3934/dcdss.2017031 |
| [21] | Richter, H., The generalized Hénon maps: Examples for higher-dimensional chaos, Int. J. Bifurcation Chaos, 12, 1371-1384, 2002 · Zbl 1044.37026 · doi:10.1142/S0218127402005121 |
| [22] | Gonchenko, S.; Gonchenko, A.; Kazakov, A.; Samylina, E., On discrete Lorenz-like attractors, Chaos, 31, 023117, 2021 · Zbl 1465.37047 · doi:10.1063/5.0037621 |
| [23] | Gonchenko, A.; Gonchenko, S. V.; Kazakov, A.; Turaev, D., Simple scenarios of onset of chaos in three-dimensional maps, Int. J. Bifurcation Chaos, 24, 1440005, 2014 · Zbl 1300.37024 · doi:10.1142/S0218127414400057 |
| [24] | Gonchenko, A.; Gonchenko, S.; Turaev, D., Doubling of invariant curves and chaos in three-dimensional diffeomorphisms, Chaos, 31, 113130, 2021 · Zbl 07871553 · doi:10.1063/5.0068692 |
| [25] | Zhang, X., Chaotic polynomial maps, Int. J. Bifurcation Chaos, 26, 1650131, 2016 · Zbl 1345.37046 · doi:10.1142/S0218127416501315 |
| [26] | Kuznetsov, Y.; Meijer, H., Numerical Bifurcation Analysis of Maps, 2019, Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1425.37002 |
| [27] | Gallas, J., Dissecting shrimps: Results for some one-dimensional physical models, Phys. A, 202, 196-223, 1994 · doi:10.1016/0378-4371(94)90174-0 |
| [28] | MacKay, R.; Tresser, C., Some flesh on the skeleton: The bifurcation structure of bimodal maps, Physica D, 27, 412-422, 1987 · Zbl 0626.58038 · doi:10.1016/0167-2789(87)90040-6 |
| [29] | Façanha, W.; Oldeman, B.; Glass, L., Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps, Phys. Lett. A, 377, 1264-1268, 2013 · Zbl 1290.37026 · doi:10.1016/j.physleta.2013.03.025 |
| [30] | Meiss, J. D.; Sander, E., Birkhoff averages and the breakdown of invariant tori in volume-preserving maps, Physica D, 428, 133048, 2021 · Zbl 1491.37076 · doi:10.1016/j.physd.2021.133048 |
| [31] | Broer, H.; Simó, C.; Vitolo, R., Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance “bubble”, Physica D, 237, 1773-1799, 2008 · Zbl 1165.37017 · doi:10.1016/j.physd.2008.01.026 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
