Document Zbl 1101.78327

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N.</a></div> <h2 class="title"> <strong>Ideal turbulence.</strong> <i>(English)</i> <a class="label nowrap" href="/1101.78327">Zbl 1101.78327</a> </h2> <div class="source"> <a href="/serials/2322" title="Journal Profile">Nonlinear Dyn.</a> <a href="/?q=in%3A181829" title="Articles in this Issue">44, No. 1-4, 15-27 (2006)</a>. </div> <div class="abstract">Summary: Ideal turbulence is a mathematical phenomenon which occurs in certain infinite-dimensional deterministic dynamical systems and implies that the attractor of a system lies off the phase space and among the attractor points there are fractal or even random functions. A mathematically rigorous definition of ideal turbulence is based on standard notions of dynamical systems theory and chaos theory. Ideal turbulence is observed in various idealized models of real distributed systems of electrodynamics, acoustics, radiophysics, etc. In systems without internal resistance, cascade processes are capable to birth structures of arbitrarily small scale and even to cause stochastization of the systems. Just these phenomena are inherent in ideal turbulence and they help to understand the mathematical scenarios for many features of real turbulence.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5067197">Cited in <strong>9</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A78A55" title="MSC2020">78A55</a> </td> <td class="space"> Technical applications of optics and electromagnetic theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37N20" title="MSC2020">37N20</a> </td> <td class="space"> Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94C05" title="MSC2020">94C05</a> </td> <td class="space"> Analytic circuit theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aboundary+value+problem">boundary value problem</a>; <a href="/?q=ut%3Adeterministic+dynamical+system">deterministic dynamical system</a>; <a href="/?q=ut%3Afractal+function">fractal function</a>; <a href="/?q=ut%3Arandom+function">random function</a>; <a href="/?q=ut%3Aspatial-temporal+chaos">spatial-temporal chaos</a>; <a href="/?q=ut%3Astochastization">stochastization</a>; <a href="/?q=ut%3Aturbulence">turbulence</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 1101.78327" data-ciurl="/ci/05067197" data-biburl="/bibtex/05067197.bib" data-amsurl="/amsrefs/05067197.bib" data-xmlurl="/xml/05067197.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05067197.pdf" title="Zbl 1101.78327 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/10.1007/s11071-006-1931-7" aria-label="DOI for “Ideal turbulence”" title="10.1007/s11071-006-1931-7">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Lukin, K. 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N., Iteration of continuous functions and dynamics of solutions for some boundary value problems, in Proceedings of ECIT-98, Annals of Mathematics Silesianae, 13, 1999, 243–255. · <a href="/0960.37014" class="nowrap">Zbl 0960.37014</a></td> </tr><tr> <td>[18]</td> <td class="space">Sharkovsky, O. M., ’Dynamical systems generated by boundary value problems. Ideal turbulence. Computer turbulence’, in Proceedings of the Ukrainian Mathematical Congress, Vol. 12, Kiev, Ukraine, 2003, pp. 125–149 (in Ukrainian). · <a href="/1100.37053" class="nowrap">Zbl 1100.37053</a></td> </tr><tr> <td>[19]</td> <td class="space">Sharkovsky, A. N., ’Difference equations and boundary value problems’, in New Progress in Difference Equations, Proceedings of ICDEA’2001, Taylor & Francis, 2004, pp. 3–22. · <a href="/1065.39040" class="nowrap">Zbl 1065.39040</a></td> </tr><tr> <td>[20]</td> <td class="space">Sharkovsky, A. N., ’Ideal turbulence: Definition’, in Proceedings of ECIT’02, Grazer Mathematishche Berichte, 346, 2004, 403–412. · <a href="/1053.37069" class="nowrap">Zbl 1053.37069</a></td> </tr><tr> <td>[21]</td> <td class="space">Sharkovsky, A. N. and Berezovsky, S. A., ’Computer turbulence’, in Proceedings of the International Conference on Self-Similar SystemsDubna, 1998, Joint Institute of Nuclear Reseach, Dubna, Russia, 1999, pp. 251–259. · <a href="/0991.65078" class="nowrap">Zbl 0991.65078</a></td> </tr><tr> <td>[22]</td> <td class="space">Sharkovsky, A. N. and Berezovsky, S. A., ’Transitions of correct–incorrect numerical calculations for solutions of some problems. Phase transitions in computer turbulence’, in Proceedings of the 2nd International Conference on Control of Oscillations and Chaos (COC’2000), Vol. 1, St. Petersburg, Russia, 2000, pp. 6–9. · <a href="/1056.37097" class="nowrap">Zbl 1056.37097</a></td> </tr><tr> <td>[23]</td> <td class="space">Sharkovsky, A. 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O., ’Dry turbulence from a time delayed Chua’s circuit’, Journal of Circuits Systems and Computers 3(2), 1993, 645–668. · <a href="https://doi.org/10.1142/S021812669300040X" class="nowrap">doi:10.1142/S021812669300040X</a></td> </tr><tr> <td>[26]</td> <td class="space">Sharkovsky, A. N., Maistrenko, Yu. L., and Romanenko, E. Yu., Difference Equations and Their Applications, Kluwer Academic, Dordrecht, The Netherlands, 1993.</td> </tr><tr> <td>[27]</td> <td class="space">Sharkovsky, A. N. and Romanenko, E. Yu., ’Problems of turbulence theory and iteration theory’, in Proceedings of ECIT-91, World Scientific, Singapore, 1992, pp. 241–252.</td> </tr><tr> <td>[28]</td> <td class="space">Sharkovsky, A. N. and Romanenko, E. Yu., ’Ideal turbulence: Attractors of deterministic systems may lie in the space of random fields’, International Journal of Bifurcation and Chaos 2(1), 1992, 31–36. · <a href="/0899.76238" class="nowrap">Zbl 0899.76238</a> · <a href="https://doi.org/10.1142/S0218127492000045" class="nowrap">doi:10.1142/S0218127492000045</a></td> </tr><tr> <td>[29]</td> <td class="space">Sharkovsky, A. N. and Romanenko, E. Yu., ’Difference equations and dynamical systems generated by some classes of boundary value problems’, in Proceedings of the Steklov Institute of Mathematics, Moscow, Vol. 244, 2004, pp. 264–279. · <a href="/1075.37032" class="nowrap">Zbl 1075.37032</a></td> </tr><tr> <td>[30]</td> <td class="space">Sharkovsky, A. N. and Romanenko, E. Yu., ’Turbulence, ideal’, in Encyclopedia of Nonlinear Science, Alwyn Scott (ed.), Routledge, New York, 2005, pp. 955–957.</td> </tr><tr> <td>[31]</td> <td class="space">Sharkovsky, A. N., Romanenko, E. Yu., and Berezovsky, S. 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G., ’Universal phenomena in solution bifurcations of some boundary value problems’, Nonlinear Mathematics and Physics 1, 1994, 147–157. · <a href="/0955.37501" class="nowrap">Zbl 0955.37501</a> · <a href="https://doi.org/10.2991/jnmp.1994.1.2.2" class="nowrap">doi:10.2991/jnmp.1994.1.2.2</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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