Document Zbl 1472.37064

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K.</a>; <a href="/authors/?q=ai%3Ahan.limin%7Chan.le%7Chan.luyi%7Chan.larry%7Chan.lifang%7Chan.liguo%7Chan.longjun%7Chan.lixia%7Chan.liang%7Chan.linghui%7Chan.lingling%7Chan.liyang%7Chan.luqing%7Chan.liming%7Chan.laiju%7Chan.lijun%7Chan.liwei%7Chan.lifeng%7Chan.liyan%7Chan.lixin%7Ckhan.lakhkar%7Chan.libin%7Chan.lili%7Ckhan.laiq%7Chan.lijia%7Chan.longsheng%7Chan.lun%7Chan.lu%7Chan.lihong%7Chan.lijuan%7Chan.liuyue%7Chan.lihui%7Chan.long%7Chan.lingxiong%7Ckhan.liyaquat%7Chan.lin%7Chan.lijie%7Chan.lei%7Chan.liubin%7Chan.lihua%7Ckhan.latifur%7Chan.liping%7Chan.lingzhan%7Chan.liubing%7Chan.longxi%7Chan.litao%7Ckhan.ladlay%7Chan.liu%7Chan.lie%7Chan.ling%7Chan.lianghao%7Chan.lefei%7Chan.lanshan%7Chan.lianfang%7Chan.luchang%7Chan.liting%7Chan.lansheng%7Chan.lining%7Chan.leitao%7Chan.luhui%7Chan.luofeng%7Chan.lengyi%7Chan.lingjuan%7Chan.liantao%7Chan.lidong%7Chan.liqin%7Chan.lixing%7Chan.lushan%7Chan.li%7Chan.liange%7Chan.lina%7Chan.lianyi%7Chan.leixin%7Chan.leng%7Chan.luxia%7Chan.linhai%7Chan.liangxiu%7Chan.liwen" title="Author Profile">Han, L.</a></div> <h2 class="title"> <strong>Linear encoding of the spatiotemporal cat.</strong> <i>(English)</i> <a class="label nowrap" href="/1472.37064">Zbl 1472.37064</a> </h2> <div class="source"> <a href="/serials/173" title="Journal Profile">Nonlinearity</a> <a href="/?q=in%3A463874" title="Articles in this Issue">34, No. 5, 2800-2836 (2021)</a>. </div> <div class="abstract">The technical novelty of the paper is a working out of a very special model of many-particle dynamics, interpreted as a discretization of a classical $d$-dimensional field theory. The main motivation of this endeavour is to describe, by means of the discrete symbolic dynamics, the spatiotemporal chaos (or turbulence) in spatially extended, strongly nonlinear field theories. To exemplify the idea, one may invoke an approximation of turbulent motions of a physical flow, in terms of coupled map lattice models (with discretized spacetime labels). It is the dynamics of finite domains that captures important small-scale spatial structures, being in turn modeled by discrete time maps (Poincaré sections of a single ‘particle’ dynamics) attached to lattice sites. It is presumed that neighboring sites are coupled, in consistency with ranslational and reflection symmetries of the problem. This particular path of reasoning is followed in the present paper by exploiting the coupled cat map lattice built from the Thom-Anosov-Arnol’d-Sinai cat maps (modeling the Hamiltonian dynamics of individual ‘particles’) at sites of a one-dimensional spatial lattice, presumed to be linearly coupled (hence ‘linear encoding’) to their nearest neighbors. The key insight is that the arising two-dimensional spatiotemporal pattern is best described by the corresponding two-dimensional spatiotemporal symbol lattice rather than by a one-dimensional temporal symbol sequence. In case of the spatiotemporal cat its every solution is uniquely encoded by a linear transformation to the corresponding finite alphabet two-dimensional symbol lattice, a spatiotemporal generalization of the linear code for temporal evolution of a cat map, introduced by <span class="zbmathjax-textit">I. Percival</span> and <span class="zbmathjax-textit">F. Vivaldi</span> [Physica D 27, 373–386 (1987; <a href="/0647.58031">Zbl 0647.58031</a>)]. It is shown that the state of the system over a finite spatiotemporal domain can be described with exponentially increasing precision by a finite pattern of symbols. A systematic, lattice Green function methodology is provided to calculate the frequency (i.e., the measure) of such states. The authors rephrase this result as follows: “local dynamics, observed through a finite spatiotemporal window, can often be thought of as a visitation sequence of a finite repertoire of finite patterns. To make statistical predictions about the system, one needs to know how often a given pattern occurs.”<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A7447">Piotr Garbaczewski (Opole)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A7353380">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A7353380">Cited in <strong>4</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A37J46" title="MSC2020">37J46</a> </td> <td class="space"> Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37J40" title="MSC2020">37J40</a> </td> <td class="space"> Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37E30" title="MSC2020">37E30</a> </td> <td class="space"> Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37D45" title="MSC2020">37D45</a> </td> <td class="space"> Strange attractors, chaotic dynamics of systems with hyperbolic behavior </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A39A23" title="MSC2020">39A23</a> </td> <td class="space"> Periodic solutions of difference equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37B10" title="MSC2020">37B10</a> </td> <td class="space"> Symbolic dynamics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aperiodic+orbits">periodic orbits</a>; <a href="/?q=ut%3Asymbolic+dynamics">symbolic dynamics</a>; <a href="/?q=ut%3Amany-particle+systems">many-particle systems</a>; <a href="/?q=ut%3Acoupled+map+lattices">coupled map lattices</a>; <a href="/?q=ut%3Achaotic+field+theory">chaotic field theory</a>; <a href="/?q=ut%3Acat+maps">cat maps</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0647.58031">Zbl 0647.58031</a></div> <div class="software"> <h3>Software:</h3><a href="/software/554">Mathematica</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1472.37064" data-ciurl="/ci/07353380" data-biburl="/bibtex/07353380.bib" data-amsurl="/amsrefs/07353380.bib" data-xmlurl="/xml/07353380.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07353380.pdf" title="Zbl 1472.37064 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1088/1361-6544/abd7c8" aria-label="DOI for “Linear encoding of the spatiotemporal cat”" title="10.1088/1361-6544/abd7c8">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1912.02940"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Akila, M.; Gutkin, B.; Braun, P.; Waltner, D.; Guhr, T., Semiclassical prediction of large spectral fluctuations in interacting kicked spin chains, Ann. 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