Bifurcations of one- and two-dimensional maps
Bifurcations of one- and two-dimensional maps
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一维和二维地图的分叉
DOI：
10.1098/rsta.1984.0020
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
D. Whitley
中科院分区：
文献类型：
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作者：
P. Holmes;D. Whitley
研究了F^e(x, y) = (y, -ex+ F (y))平面映射的两参数族的定性动力学，其中F:R - >r是具有单临界点和负Schwarzian导数的C3映射。这种地图的原型是族f(y) = u -y2或（在不同的坐标下）f(y) = Ay(1 -y)，在这种情况下f ^e是Henon地图。映射Fe有恒定的雅可比行列式e，当e - >0时，分解成族f^。这种一维族的行为已经被很好地理解，我们能够利用它们的分岔结构和它们的非游走集上的信息得到F/ue的局部分岔和全局分岔的结果，对于小e，我们能够将这些结果推广到保域族F/u。1，从而得到(/u, e)-平面上的（部分）分岔集。在我们的结论中，我们发现周期轨道的分岔序列在一维映射的Sarkovskii定理和揉合理论的限制下，在二维族中有很大的不同。特别是，某些出现在一维序列末端的周期轨道出现在保面积序列的起始位置，在e = 0和e = 1之间的（/u, e）参数平面上，无限多族的鞍节点和倍周期分岔曲线相互交叉。我们通过研究F/u的同斜分岔（稳定流形和不稳定流形的切线）得到了这些结果。E和累积在它们上的周期分岔的相关序列的值。我们用一些保持方向的Henon映射的数值计算来说明我们的结果。
We study the qualitative dynamics of two-parameter families of planar maps of the form F^e(x, y) = (y, -ex+f(y)), where f :R -> R is a C3 map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f(y) = u —y2 or (in different coordinates) f(y) = Ay(1 —y), in which case F^e is the Henon map. The maps Fe have constant Jacobian determinant e and, as e -> 0, collapse to the family f^. The behaviour of such one-dimensional families is quite well understood, and we are able to use their bifurcation structures and information on their non-wandering sets to obtain results on both local and global bifurcations of F/ue, for small e. Moreover, we are able to extend these results to the area preserving family F/u.1, thereby obtaining (partial) bifurcation sets in the (/u, e)-plane. Among our conclusions we find that the bifurcation sequence for periodic orbits, which is restricted by Sarkovskii’s theorem and the kneading theory for one-dimensional maps, is quite different for two-dimensional families. In particular, certain periodic orbits that appear at the end of the one-dimensional sequence appear at the beginning of the area preserving sequence, and infinitely many families of saddle node and period doubling bifurcation curves cross each other in the (/u, e) -parameter plane between e = 0 and e = 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F/u.e and of the associated sequences of periodic bifurcations that accumulate on them. We illustrate our results with some numerical computations for the orientation-preserving Henon map.