Compound windows of the Hénon-map. (English) Zbl 1155.37030

Lorenz, Edward N.

Physica D 237, No. 13, 1689-1704 (2008).

For the standard 2-parameter 2nd order Henon map shapes and locations of periodic windows in the bifurcation diagram are studied numerically by means of a random searching procedure. To study the stability of solutions the author introduces a functional whose values smaller than one on modulus indicate the stability of the portion of a window where a fundamental period prevails.

Reviewer: Michael L. Blank (Moskva)

MSC:

37E99	Low-dimensional dynamical systems
37C99	Smooth dynamical systems: general theory
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

Henon map; bifurcation diagram; periodic window

References:

[1]	Benedicks, M.; Carleson, L., The dynamics of the Hénon map, Ann. math., 133, 73-169, (1991) · Zbl 0724.58042
[2]	Carcassès, J.P.; Mira, C.; Bosch, M.; Simó, C.; Tatjer, J.C., Crossroad area—spring area transition. (1) parameter plane representation, Internat J. bifur. chaos, 1, 183-196, (1991) · Zbl 0758.58025
[3]	Collet, P.; Eckmann, J.-P., On the abundance of aperiodic behavior for maps of the interval, Commun. math. phys., 73, 115-160, (1980) · Zbl 0441.58011
[4]	Curry, J.H., On the Hénon transformation, Comm. math. phys., 68, 129-140, (1979) · Zbl 0414.58024
[5]	Feigenbaum, M., Quantitative universality for a class of nonlinear transformations, J. stat. phys., 19, 25-52, (1978) · Zbl 0509.58037
[6]	Feit, S.B., Characteristic exponents and strange attractors, Comm. math. phys., 61, 249-260, (1978) · Zbl 0399.65091
[7]	Guckenheimber, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer New York
[8]	Hénon, M., A two-dimensional mapping with a strange attractor, Commun. math. phys., 50, 69-77, (1976) · Zbl 0576.58018
[9]	Hitzl, D.L.; Zele, F., An exploration of the Hénon map, Physica D, 14, 305-326, (1985) · Zbl 0602.58025
[10]	Hunt, B.H.; Kennedy, J.A.; Li, T.-Y.; Nusse, H.E., SLYRB measures: natural invariant measures for chaotic systems, Physica D, 170, 50-71, (2002) · Zbl 1098.37517
[11]	Jakobson, M.V., Absolutely continuous measures for one-parameter families of one-dimensional maps, Comm. math. phys., 81, 39-88, (1981) · Zbl 0497.58017
[12]	Julia, G., Mémoire sur l’itération des fonctions rationelles, J. math. pure appl., 4, 47-245, (1918) · JFM 46.0520.06
[13]	Lauwerier, H.A., ()
[14]	Lorenz, E.N., The problem of deducing the climate from the governing equations, Tellus, 16, 1-11, (1964)
[15]	Lozi, R., Un attracteur étrange (?) due type attracteur de Hénon, J. phys. (Paris), 39, 9-10, (1978)
[16]	May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467, (1976) · Zbl 1369.37088
[17]	Mira, C., Chaotic dynamics. from the one-dimensional endomorphism to the two-dimensional diffeomorphism, (1987), World Scientific Singapore · Zbl 0641.58002
[18]	Mira, C.; Carcassès, J.P.; Bosch, M.; Simó, C.; Tatjer, J.C., Crossroad area—spring area transition. (2) foliated parameter representation, Internat J. bifur. chaos, 1, 339-348, (1991) · Zbl 0876.58035
[19]	Mira, C.; Carcassès, J.P., On the “crossroad area—saddle area“ and “crossroad area—spring area” transitions, Internat. J. bifur. chaos, 1, 641-655, (1991) · Zbl 0874.58060
[20]	Olsen, L.F.; Degn, H., Chaos in biological systems, Q. rev. biophys., 18, 165-225, (1985)
[21]	Simó, C., On the Hénon – pomeau attractor, J. stat. phys., 21, 465-494, (1979)
[22]	Gallas, J.A.C., Structure of the parameter space of the Hénon map, Phys. rev. let., 70, 2714, (1993)
[23]	Gallas, J.A.C., Dissecting shrimps: results for some one-dimensional physical systems, Physica A, 202, 196, (1994)
[24]	Bonatto, C.; Gallas, J.A.C.; Ueda, Y., Chaotic phase similarities and recurrences in a damped-driven duffling oscillator, Phys. rev. E, 77, 026217, (2008)
[25]	Hunt, B.R..; Gallas, J.A.C.; Grebogi, C.; Yorke, J.A.; Koçak, H., Bifurcation rigidity, Physica D, 129, 35, (1999) · Zbl 0951.37010