Yang Hong-liu, Radons Günter
Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Jan;73(1 Pt 2):016208. doi: 10.1103/PhysRevE.73.016208. Epub 2006 Jan 12.
In our previous study of hydrodynamic Lyapunov modes (HLMs) in coupled map lattices, we found that there are two classes of systems with different lambda-k dispersion relations. For coupled circle maps we found the quadratic dispersion relations lambda approximately k2 and lambda approximately k for coupled standard maps. Here, we carry out further numerical experiments to investigate the dynamic Lyapunov vector (LV) structure factor which can provide additional information on the Lyapunov vector dynamics. The dynamic LV structure factor of coupled circle maps is found to have a single peak at omega=0 and can be well approximated by a single Lorentzian curve. This implies that the hydrodynamic Lyapunov modes in coupled circle maps are nonpropagating and show only diffusive motion. In contrast, the dynamic LV structure factor of coupled standard maps possesses two visible sharp peaks located symmetrically at +/- omega. The spectrum can be well approximated by the superposition of three Lorentzian curves centered at omega=0 and +/-omegau, respectively. In addition, the omega-k dispersion relation takes the form omegau=cuk for k --> 2pi/L. These facts suggest that the hydrodynamic Lyapunov modes in coupled standard maps are propagating. The HLMs in the two classes of systems are shown to have different dynamical behavior besides their difference in spatial structure. Moreover, our simulations demonstrate that adding damping to coupled standard maps turns the propagating modes into diffusive ones alongside a change of the lambda-k dispersion relation from lambda approximately k to lambda approximately k2. In cases of weak damping, there is a crossover in the dynamic LV structure factors; i.e., the spectra with smaller k are akin to those of coupled circle maps while the spectra with larger k are similar to those of coupled standard maps.
在我们之前对耦合映射格点中的流体动力学李雅普诺夫模式（HLMs）的研究中，我们发现有两类系统具有不同的λ - k色散关系。对于耦合圆映射，我们发现了二次色散关系λ≈k²，而对于耦合标准映射则是λ≈k。在这里，我们进行进一步的数值实验，以研究动态李雅普诺夫向量（LV）结构因子，它可以提供关于李雅普诺夫向量动力学的额外信息。我们发现耦合圆映射的动态LV结构因子在ω = 0处有一个单峰，并且可以很好地用单个洛伦兹曲线近似。这意味着耦合圆映射中的流体动力学李雅普诺夫模式是不传播的，仅表现出扩散运动。相比之下，耦合标准映射的动态LV结构因子在±ω处对称地有两个明显的尖峰。该频谱可以很好地用分别以ω = 0和±ω₀为中心的三个洛伦兹曲线的叠加来近似。此外，对于k→2π/L，ω - k色散关系具有ω₀ = cuk的形式。这些事实表明耦合标准映射中的流体动力学李雅普诺夫模式是传播的。除了空间结构上的差异外，这两类系统中的HLMs还表现出不同的动力学行为。此外，我们的模拟表明，给耦合标准映射添加阻尼会使传播模式变为扩散模式，同时λ - k色散关系从λ≈k变为λ≈k²。在弱阻尼情况下，动态LV结构因子存在交叉；即，较小k的频谱类似于耦合圆映射的频谱，而较大k的频谱类似于耦合标准映射的频谱。
