| 研究実績の概要 |
Special subsets of orbits, such as the periodic, homoclinic, and closed orbits, play central roles in the semiclassical quantization of the quantum chaotic systems. For example, in the Gutzwiller's trace formula, the energy density of state is approximated by an interference sum of the unstable periodic orbits arising from the classical limit of the quantum system. The contribution of each periodic orbit is determined by its phase and amplitude, which are governed by the classical functions and the stability exponents of orbit, respectively. Short periodic orbits give the spectra to low resolutions, and longer periodic orbits give the details of the spectra in higher resolutions. However, while the short periodic orbits may be easy to calculate, due to the exponential instability of chaotic systems, the numerical construction of long periodic orbits are intrinsically difficult. In my PhD thesis, I have developed a theoretical framework to replace the unstable periodic orbits by homoclinic orbits. While the former are difficult to find, the latter arise from intersections of stable and unstable manifolds, therefore can be identified in systematic ways feasible for numerical implementations. A major limitation of the framework is that it is based on two-dimensional systems. During my previous year as the JSPS fellow, we have developed novel phase-space techniques exploiting the properties of invariant manifolds in higher dimensions and successfully generalized the old framework to multidimensional systems.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
Up to this point, we have developed a theoretical scheme to express the important properties of any phase-space trajectories in terms of the homoclinic orbits. The homoclinic orbits are established as the skeletal structures of the dynamics, which can be used to calculate all other kinds of orbits in the chaotic systems. The current framework is based on Markov partitions of the phase space and the corresponding symbolic dynamics arising from the partitions. Each orbit is represented by a bi-infinite string of digits which registers its itinerary in terms of the Markov partitions under the dynamics. The current status of our work is summarized in arXiv:2009.12224 [nlin.CD]. However, the current approach is based on the assumptions that Markov partitions exist for systems that are "sufficiently chaotic". For non-uniformly hyperbolic systems, recent advances by Sarig [O. M. Sarig. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc. 26(2) (2013), 341-426] and Ben Ovadia [S. Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn. 13 (2018), 43-113] established the existence of Markov partitions, which then grantees the applicability of our approach in such systems. However, for systems that are less chaotic then non-uniformly hyperbolic but still chaotic in every visible sense, such as the standard map with large kicking strengths, it is intuitive that our theory should work but the existence of Markov partitions is not proven or even known to exist.
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| 今後の研究の推進方策 |
Our plan for this year is to study the possibility of constructing an analog of the Markov partitions for mixed systems displaying large portions of chaotic seas and small partitions of stability islands. This scheme, once succeeded, will enable the construction of exact or approximate symbolic dynamics for the chaotic trajectories in mixed systems, which will represent a major breakthrough in the study of generic chaotic dynamics and enable semiclassical calculations of a wide class of the quantum systems which were not accessible before.
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