Geodesic
Lorenz- and Shilnikov-Shape Attractors
Russian journal of nonlinear dynamics, Tome 17 (2021) no. 2, pp. 165-174.

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We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble the well-known discrete Lorenz-like attractors and those which are similar to the discrete Shilnikov attractors. We describe and illustrate the scenarios of occurrence of chaotic attractors of both types.
Keywords: strange attractor, discrete Lorenz attractor, hyperchaos, discrete Shilnikov attractor, two-dimensional endomorphism. 
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E. Kuryzhov; E. Karatetskaia; D. Mints. Lorenz- and Shilnikov-Shape Attractors. Russian journal of nonlinear dynamics, Tome 17 (2021) no. 2, pp. 165-174. http://geodesic.mathdoc.fr/item/ND_2021_17_2_a2/

[1] Biragov, V. S., Ovsyannikov, I. M., and Turaev, D. V., “A Study of One Endomorphism of a Plane”, Methods of Qualitative Theory and Theory of Bifurcations, ed. E. A. Leontovich-Andronova, Gorky Gos. Univ., Gorky, 1988, 72–86 (Russian) | MR

[2] Gonchenko, S. V., Ovsyannikov, I. I., Simó, C., and Turaev, D., “Three-Dimensional Hénon-Like Maps and Wild Lorenz-Like Attractors”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15:11 (2005), 3493–3508 | DOI | MR | Zbl

[3] Gonchenko, A. S., Gonchenko, S. V., and Shilnikov, L. P., “Towards Scenarios of Chaos Appearance in Three-Dimensional Maps”, Nelin. Dinam., 8:1 (2012), 3–28 (Russian) | DOI

[4] Gonchenko, A. S., Gonchenko, S. V., Kazakov, A. O., and Turaev, D. V., “Simple Scenarios of Onset of Chaos in Three-Dimensional Maps”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24:8 (2014), 1440005, 25 pp. | DOI | MR | Zbl

[5] Gonchenko, A. S. and Gonchenko, S. V., “Variety of Strange Pseudohyperbolic Attractors in Three-Dimensional Generalized Hénon Maps”, Phys. D, 337 (2016), 43–57 | DOI | MR | Zbl

[6] Gonchenko, A. S., Gonchenko, S. V., and Kazakov, A. O., “Richness of Chaotic Dynamics in the Nonholonomic Model of Celtic Stone”, Regul. Chaotic Dyn., 18:5 (2013), 521–538 | DOI | MR | Zbl

[7] Izv. Vyssh. Uchebn. Zaved. Radiofizika, 62:5 (2019), 412–428 (Russian) | DOI

[8] Eilertsen, J. S. and Magnan, J. F., “On the Chaotic Dynamics Associated with the Center Manifold Equations of Double-Diffusive Convection near a Codimension-Four Bifurcation Point at Moderate Thermal Rayleigh Number”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28:8 (2018), 1850094, 24 pp. | DOI | MR | Zbl

[9] Eilertsen, J. S. and Magnan, J. F., “Asymptotically Exact Codimension-Four Dynamics and Bifurcations in Two-Dimensional Thermosolutal Convection at High Thermal Rayleigh Number: Chaos from a Quasi-Periodic Homoclinic Explosion and Quasi-Periodic Intermittency”, Phys. D, 382/383 (2018), 1–21 | DOI | MR | Zbl

[10] Kuznetsov, Yu. A., Meijer, H. G. E., and van Veen, L., “The Fold-Flip Bifurcation”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14:7 (2004), 2253–2282 | DOI | MR | Zbl

[11] Ovsyannikov, I., Global and Local Bifurcations, Three-Dimensional Hénon Maps and Discrete Lorenz Attractors, 2021, arXiv: 2104.01262 [math.DS] | Zbl

[12] Selecta Math. Soviet., 10:1 (1991), 43–53 | MR | MR | Zbl

[13] Garashchuk, I. R., Sinelshchikov, D. I., Kazakov, A. O., and Kudryashov, N. A., “Hyperchaos and Multistability in the Model of Two Interacting Microbubble Contrast Agents”, Chaos, 29:6 (2019), 063131, 16 pp. | DOI | Zbl

[14] Rössler, O. E., “An Equation for Hyperchaos”, Phys. Lett. A, 71:2–3 (1979), 155–157 | DOI | MR

[15] Harrison, M. A. and Lai, Y.-Ch., “Route to High-Dimensional Chaos”, Phys. Rev. E, 59:4 (1999), R3799–R3802 | DOI | MR

[16] Shykhmamedov, A., Karatetskaia, E., Kazakov, A., and Stankevich, N., Hyperchaotic Attractors of Three-Dimensional Maps and Scenarios of Their Appearance, 2020, arXiv: 2012.05099 [math.DS]

[17] Stankevich, N., Kazakov, A., and Gonchenko, S., “Scenarios of Hyperchaos Occurrence in 4D Rössler System”, Chaos, 30:12 (2020), 123129, 16 pp. | DOI | MR | Zbl

[18] Karatetskaia, E., Shykhmamedov, A., and Kazakov, A., “Shilnikov Attractors in Three-Dimensional Orientation-Reversing Maps”, Chaos, 31:1 (2021), 011102, 11 pp. | DOI | MR | Zbl

[19] Gonchenko, S., Gonchenko, A., Kazakov, A., and Samylina, E., “On Discrete Lorenz-Like Attractors”, Chaos, 31:2 (2021), 023117, 20 pp. | DOI | MR | Zbl

[20] Stankevich, N., Kuznetsov, A., Popova, E., and Seleznev, E., “Chaos and Hyperchaos via Secondary Neimark – Sacker Bifurcation in a Model of Radiophysical Generator”, Nonlinear Dynam., 97:4 (2019), 2355–2370 | DOI | Zbl