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从准周期进入混沌的研究
【作者】 史朋亮；
【导师】 胡岗；
【作者基本信息】 北京师范大学 ， 物理学理论物理， 2001， 博士
【摘要】 在一个自治的耦合延时抛物映象体系中,发现从三频准周期通向混沌的道路上,二频准周期环面的锁频呈现Arnold舌头和魔鬼阶梯结构；我们数值上发现三频准周期可以稳定存在于连续的参数区域。在准周期驱动体系中,从准周期通过奇怪非混沌吸引子进入混沌的道路成为近年研究的热点。基于前人对奇怪非混沌吸引子奇怪性的认识,我们简化了相敏感指数定义,利用它研究了SNA产生时的临界行为和倍环面分岔终结进入SNA的过程,并且都得到了标度律。整个论文研究的重点是第三章耦合抛物映象中有关混沌和非混沌吸引子的数值模拟问题。应用双精度和高精度数值模拟的对比,在准周期驱动抛物映象发现了类非混沌奇怪吸引子。该吸引子在低精度时表现为为奇怪吸引子而高精度时变为光滑的准周期环。进而,我们把体系中响应系统反馈给驱动项,构造一个自治的耦合映象系统,发现了类混沌奇怪吸引子,也有类似的行为。此时类混沌奇怪吸引子在低精度时表现为奇怪吸引子,最大Lyapunov指数(简称李指数)可正可负；在高精度下二者皆变为光滑的准周期环。这种类非混沌及类混沌吸引子的行为源于系统虽有负的第二大李指数,但存在具有极大的局域放大效果的局域正李指数。藉此我们深入探讨了非线性研究中的一个重要问题：数值模拟的可信性问题。Shadow引理保证了严格双曲体系中任何伪轨道附近存在真实轨道。但众多非线性体系都不严格双曲。我们发现,在非严格双曲体系中,数值模拟的精度不能低于体系的最大伸缩量的倒数；否则,由数值模拟得到的伪轨道就不能反映系统的真实轨道,从而严重影响我们对体系真实行为的认识。这里体系的最大伸缩量与系统的局部李指数有着密切关系。我们对混沌和非混沌体系都进行了研究,得到了统一的结论。特别指出的是这一研究具有较强的实验研究价值。在实验中不可避免的存在各种噪声,在局部李指数具有较强局部放大的非混沌非奇怪吸引子系统,噪声的存在会使系统表现出明显的类似混沌和奇怪吸引子的性质,这使非混沌系统中类混沌和奇怪吸引子的行为在实验和实际生活中能较普遍的观察到。
【Abstract】 The route from quasiperiodicity to chaos is a hot topic in nonlinear science research. In my work several different dynamical systems including autonomous coupled maps and quasiperiodically driven maps are investi-gated. Around Strange Nonchaotic Attractor(SNA) we discuss some scaling relations of the critical behaviors. Another fundamental problem of the nu-merical simulation about Shadow Lemma in nonlinear science is also anal-ysed numerically. We managed to answer the problem how high computing precision should be used for giving correct dynamics behavior in nonlinear systems.Devil’s staircases and Arnold’s tongues are commonly seen in many dy-namical systems. We find that in our autonomous maps these tongues se-quences can appear in the locking process between3-torus and2-torus. The three dimensional quasiperiodic attractors in the system are stable and they can appear in a continuous region in the parameter space.Strangeness is one of the main characters of SNA. We defined a Phase Sensitivity Exponent (PSE) quantity to measure the strange geometry struc-ture, developing the result of A. Pikovsky. Using the PSE we get the scaling law near the critical point of the birth of SNA and the scaling of the torus doubling terminal points in quasiperiodically driven maps.The Shadow Lemma guarantees a good property for the hyperbolic sys-tems that there always exists a true trajectory near any pseudo-trajectory if the computing error is sufficiently low. However, there are only very few ex-amples of robust hyperbolic systems, such as the Smale-Williams’solenoid or Plykin’s attractor. Most of nonlinear systems are not hyperbolic. Therefore every scientists must face such a question:Whether the computer-generated trajectories stand for true attractors of the systems? Actually, if the comput-ing precision is not enough high, the pseudo-trajectories computer-generated will give a false picture of the system and some wrong conclusions may be made. How high precision we should apply in a given system is the key problem. Based on the investigation of coupled logistic maps, we get a clear conclusion:if a nonchaotic system has a maximal expanding ability of10β and the distinction in the system dynamics is10-γ, then the numerical sim-ulation should have a roundoff smaller than10-α, where α=β+γ. The maximal expanding ability can be get from the product of the positive local Lyapunov exponent and its finite time length. For too large expanding rate and low computing precision the numerical simulation can generate strange or chaotic behavior for nonstrange and nonchaotic systems. We call such attractors SNA-like or Chaos-like Attractors. For a chaotic system, if it has a maximal local contracting ability of10-β, and the attractor, which has the size of10-γ, will collapse to periodic pseudo-trajectories if the roundoff of the simulation10-α is larger than10-(β+γ); i. e., only when α>β+γ, the true picture of chaotic attractor can be achieved. We expect that the above understanding may be applied in systematically determining the compute precision for different nonlinear systems.