Resonant collisions in four-dimensional reversible maps: A description of scenarios
- 作者：Lahiri ; Avijit ; Bhowal ; Ajanta ; Roy ; Tarun K.
- 关键词：05.45.+b ; 03.20.+i ; Reversible maps ; Bifurcation ; Resonant collisions of multipliers ; Secondary bifurcation ; Reversible Hopf bifurcation
- 刊名：Physica D
- 出版年：1998
- 期刊代码：55_01672789
- 类别：ph
- 出版时间：January 15, 1998
- 卷：112
- 期：1-2
- 页码：95-116
- 文件大小：1.05 M
摘要
We define a resonant collision of order k (≥ 1) in a family of four-dimensional (4D) reversible maps. For any specified k, the bifurcation scenario is the collection of the different possible types of bifurcation of a symmetric fixed point that may be encountered through various choices of parameters describing the family of maps under consideration. We adopt a perturbative approach, coupled with numerical iterations around orbits obtained perturbatively, to explore phase space structures in the immediate vicinity of a resonant collision and thereby to obtain a description of the possible scenarios for different values of k. The phase space structures typically involve bifurcating periodic orbits, families of invariant curves, and tori, and present interesting possibilities, especially around the ‘secondary bifurcations’ of the periodic orbits. Based on the results of the perturbative and numerical approach we conjecture that three distinct scenarios are involved for the cases k = 2, 3, 4, respectively, while there exists a fourth distinctive scenario common to all k > 4, and we present what we believe to be a reasonably exhaustive description of these scenarios. The case k = 1 involves bifurcations of fixed points rather than of periodic orbits, and has been investigated numerically in an earlier paper.