For citation:
Zhusubaliev Z. T., Rudakov V. N., Soukhoterin E. A. On dynamics of the Henon map. Izvestiya VUZ. Applied Nonlinear Dynamics, 2000, vol. 8, iss. 3, pp. 3-16. DOI: 10.18500/0869-6632-2000-8-3-3-16
On dynamics of the Henon map
Division of the parameter plane for the two—dimensional Henon mapping into domains of periodic and chaotic oscillations is studied numerically and analytically. Regularities in the occurrence of different motions and bifurcational transitions are analyzed. It is shown that there are domains in the plane of parameters, where non—uniqueness of motions exists. This may lead to abrupt changes of the character of the dynamics under variation in the parameters, that is, to а sudden transition from one stable cycle to another or to chaotization of the oscillations.
- Henon M. A two-dimensional mapping with a strange attractor. Commun.Math. Phys. 1976;50:69–77. DOI: 10.1007/BF01608556.
- Hitzl DL, Zele Е. An exploration of the Henon attractors. J.Stat. Phys. 1981;26(4):683-695.
- Lichtenberg AJ, Lieberman MA. Regular and Stochastic Motion. N.Y.: Springer; 1983. 499p. DOI: 10.1007/978-1-4757-4257-2.
- Berge P, Pomeau Y, Vidal C. Order within Chaos. Weinheim: Wiley;1987. 329 p.
- Moon FC. Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. Weinheim: Wiley;1987. 309 p.
- Anishchenko VS. Complex Oscillations in Simple Systems. M.: Nauka;1990. 312 p. (in Russian).
- Dmitriev AS, Kislov VYa. Stochastic Oscillations in Radiophysics and Electronics. М.: Nauka;1989. 280 p.
- Dmitriev AS, Starkov SO, Shirokov ME. Structure of periodic orbits of a chaotic self-oscillating system described by second-order difference equations. J. Commun. Technol. Electron. 1994;39(8-9):1392-1400. (in Russian).
- Anishchenko V.S., Vadivasova T.E., Strelkova G.I., Kopeikin A.S. Chaotic attractors оf two-dimensional invertible maps. Discrete Dynamics in Nature and Society. 1998;2(4):249-256. DOI: 10.1155/S1026022698000223.
- Anishchenko VS, Strelkova GI. Irregular attractors. Discrete Dynamics in Nature and Society. 1998;2(1):53-72. DOI:10.1155/S1026022698000041.
- Zhusubaliyev ZhT, Rudakov VN, Soukhoterin EA, Mosekilde Е. Bifurcation analysis оf the Henon mар. Discrete Dynamics in Nature and Society. 2000;5(3):203-221. DOI: 10.1155/S1026022600000534.
- Baushev VS, Zhusubaliev ZhT. On non-deterministic modes of operation of the voltage stabiliser with pulse-width regulation. Electricity. 1992;8:47-53. (in Russian).
- Baushev VS, Zhusubaliev ZhT, Mikhalchenko S.G. Stochasticity in the dynamics of the voltage stabiliser with pulse-width regulation. Electricity. 1996;3:69-75. (in Russian).
- Zhusubaliev ZhT. To the study of chaotic modes of a voltage transformer with pulse-width modulation. Electricity. 1997;6:40-46. (in Russian).
- Zhusubaliev ZhT. Bifurcations and chaotic movements in the dynamics of relay automatic control systems. In: Collection of materials 3rd International Conference “Recognisation-97”. Kursk;1997. P. 25. (in Russian).
- Neymark YuI, Landa PS. Stochastic and chaotic fluctuations. Moscow: Nauka; 1987. 422 p.(In Russian).
- Butenin NV, Neimark YuI, Fufaev NA. Introduction to the theory of non-linear oscillations. M.: Nauka; 1987. 324 (In Russian).
- Mira C, Carcasses J, Millerioux С, Gardini L. Рlane foliation of two-dimensional noninvertibie maps. Int. J. Bifurc. and Chaos. 1996;6(8):1439-1462. DOI: 10.1142/S0218127496000850.
- Kuznetsov АP, Kuznetsov SP, Sataev IR. Variety of types оf critical behavior and multistability in period-doubling systems with unidirectional coupling near the onset оf chaos. Int. J. Bifurc. and Chaos. 1993;3(1):140-152. DOI:10.1142/S021812749300009X.
- Barfred M, Mosekilde E, Holstein-Rathlou N—H. Bifurcation analysis of nephron pressure and flow regulation. Chaos. 1996;6(3):280 -287. DOI: 10.1063/1.166175.
- Mosekilde Е. Торics in Nonlinear Dynamics: Applications to Physics, Biology and Economic Systems. Singapore: World Scientific; 1996. 380 p.
- Gallas J. Dissecting shrimps: results for some one-dimensional physical models. Physica А. 1994;202(1-2):196-223. DOI: 10.1016/0378-4371(94)90174-0.
- 197 reads