Department of Mathematics, University of Portsmouth, Portsmouth PO1 3HF, United Kingdom.
Chaos. 2011 Jun;21(2):023132. doi: 10.1063/1.3594577.
We consider the damped and driven dynamics of two interacting particles evolving in a symmetric and spatially periodic potential. The latter is exerted to a time-periodic modulation of its inclination. Our interest is twofold: First, we deal with the issue of chaotic motion in the higher-dimensional phase space. To this end, a homoclinic Melnikov analysis is utilised assuring the presence of transverse homoclinic orbits and homoclinic bifurcations for weak coupling allowing also for the emergence of hyperchaos. In contrast, we also prove that the time evolution of the two coupled particles attains a completely synchronised (chaotic) state for strong enough coupling between them. The resulting "freezing of dimensionality" rules out the occurrence of hyperchaos. Second, we address coherent collective particle transport provided by regular periodic motion. A subharmonic Melnikov analysis is utilised to investigate persistence of periodic orbits. For directed particle transport mediated by rotating periodic motion, we present exact results regarding the collective character of the running solutions entailing the emergence of a current. We show that coordinated energy exchange between the particles takes place in such a manner that they are enabled to overcome--one particle followed by the other--consecutive barriers of the periodic potential resulting in collective directed motion.
我们考虑了在对称且空间周期性势中相互作用的两个粒子的阻尼和驱动动力学。后者受到其倾斜度的周期性调制。我们的兴趣有两个方面：首先，我们处理了高维相空间中混沌运动的问题。为此，利用同宿梅尔尼科夫分析确保存在横向同宿轨道和弱耦合的同宿分叉，也允许出现超混沌。相比之下，我们还证明了对于两个耦合粒子之间的强耦合，它们的时间演化可以达到完全同步（混沌）状态。由此产生的“冻结维度”排除了超混沌的发生。其次，我们研究了由规则周期性运动提供的相干集体粒子输运。利用次谐波梅尔尼科夫分析来研究周期轨道的持续存在。对于由旋转周期性运动介导的定向粒子输运，我们给出了关于运行解的集体特征的精确结果，这些解涉及电流的出现。我们表明，粒子之间的协调能量交换以这样一种方式发生，即它们能够克服周期性势的连续势垒，从而导致集体定向运动。一个粒子紧随其后），导致集体定向运动。
