THE SHADOW-CURVES OF THE ORBIT DIAGRAM PERMEATE THE BIFURCATION DIAGRAM, TOO

被引:4
作者
Ross, Chip [1 ]
Odell, Meredith [1 ]
Cremer, Sarah [1 ]
机构
[1] Bates Coll, Dept Math, Lewiston, ME 04240 USA
来源
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS | 2009年 / 19卷 / 09期
关键词
Myrberg map; orbit diagram; bifurcation diagram; attracting; repelling; and indifferent periodic point; preperiodic point; Q-curve; P-curve; Misiurewicz point;
D O I
10.1142/S0218127409024621
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The "Q-curves" Q(1)(c) = c, Q(2)(c) = c(2) + c,..., Q(n)(c) = (Q(n-1)(c))(2) + c = f(c)(n) (0) have long been observed and studied as the shadowy curves which appear illusively - not explicitly drawn in the familiar orbit diagram of Myrberg's map f(c)(x) = x(2) + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x(2) + c as well - by "bifurcation diagram" we mean the collection of all periodic points of fc (attracting, indifferent and repelling) - these collections form what we call "P-curves". We show Q-curves and P-curves intersect in one of two ways: At a superattracting periodic point on a P-curve, the infinite family of Q-curves which intersect there are all tangent to the P-curve. At a Misiurewicz point, no tangencies occur at these intersections; the slope of the P-curve is the fixed point of a linear system whose iterates give the slopes of the Q-curves. We also introduce some new phenomena associated with c sin x illustrating briefly how its two different families of Q-curves interact with P-curves. Our algorithm for finding and plotting all periodic points (up to any reasonable period) in the bifurcation diagram is reviewed in an Appendix.
引用
收藏
页码:3017 / 3031
页数:15
相关论文
共 12 条
  • [1] ALLIGOOD K, 1997, CHAOS INTRO DYNAMICA, P133
  • [2] [Anonymous], 1959, ANN ACAD SCI FENN-M
  • [3] [Anonymous], ANN ACAD SCI FENNI A
  • [4] CARLESON L, 1993, COMPLEX DYNAMICS, P133
  • [5] DEVANEY RL, 1989, INTRO CHAOTIC DYNAMI, P82
  • [6] SIMPLE MATHEMATICAL-MODELS WITH VERY COMPLICATED DYNAMICS
    MAY, RM
    [J]. NATURE, 1976, 261 (5560) : 459 - 467
  • [7] Mira C., 1987, Chaotic dynamics from the one-dimensional endomorphism to the two- dimensional diffeomorphism
  • [8] Myrberg P. J., 1963, Ann. Acad. Sci. Fenn., Ser. A. I., V336, P1
  • [9] The road to chaos is filled with polynomial curves
    Neidinger, RD
    Annen, RJ
    [J]. AMERICAN MATHEMATICAL MONTHLY, 1996, 103 (08) : 640 - 653
  • [10] PEITGEN HO, 1992, CHAOS FRACTALS NEW F, P587