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The <strong>and -operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell" role="cell">The <strong>or</strong>-operator <strong>|</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3AEur%2A+J%2A+Mat%2A+Soc%2A+cc%3A14">so:Eur* J* Mat* Soc* cc:14</a></span> <span class="search-explanation" role="cell" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%27%27Partial+diff%2A+eq%2A%27%27+%21elliptic">''Partial diff* eq*'' !elliptic</a></span> <span class="search-explanation" role="cell" role="cell">The <strong>not</strong> -operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to <u>b</u>ooks; alternatively: <u>j</u> for <strong><u>j</u>ournal articles</strong>, <u>a</u> for <strong>book <u>a</u>rticles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000+-+2015+cc%3A%2894A+%7C+11T%29">py: 2000 - 2015 cc:(94A | 11T)</a></span> <span class="search-explanation" role="cell" role="cell">Number <strong>ranges</strong> when searching for <strong><u>p</u>ublication <u>y</u>ear</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href='https://en.wikipedia.org/wiki/ISO_639-1'>ISO 639 - 1</a> (opens in new tab) language codes can also be used.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=st%3A+c+r+s">st: c r s</a></span> <span class="search-explanation" role="cell" role="cell">Find documents that are <strong><u>c</u>ited</strong>, have <strong><u>r</u>eferences</strong> and are from a <strong><u>s</u>ingle author</strong>.</span> </div> </div> <p style="margin-bottom:0.5cm;"></p> <h2>Fields</h2> <div id="help-fields" role="table"> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ab</strong> </span> <span class="search-explanation" role="cell" role="cell"> Text from the summary or review (for phrases use “...”) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>an</strong> </span> <span class="search-explanation" role="cell" role="cell"> zbMATH ID, i.e.: preliminary ID, Zbl number, JFM number, ERAM number </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>any</strong> </span> <span class="search-explanation" role="cell" role="cell"> Includes ab, au, cc, en, rv, so, ti, ut </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>arxiv</strong> </span> <span class="search-explanation" role="cell" role="cell"> arXiv preprint number </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>au</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name(s) of the contributor(s) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>br</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of a person with biographic references (to find documents about the life or work) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>cc</strong> </span> <span class="search-explanation" role="cell" role="cell"> Code from the Mathematics Subject Classification (prefix with <strong>*</strong> to search only primary MSC) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ci</strong> </span> <span class="search-explanation" role="cell" role="cell"> zbMATH ID of a document cited in summary or review </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>db</strong> </span> <span class="search-explanation" role="cell" role="cell"> Database: documents in Zentralblatt f眉r Mathematik/zbMATH Open (<em>db:Zbl</em>), Jahrbuch 眉ber die Fortschritte der Mathematik (<em>db:JFM</em>), Crelle's Journal (<em>db:eram</em>), arXiv (<em>db:arxiv</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>dt</strong> </span> <span class="search-explanation" role="cell" role="cell"> Type of the document: journal article (<em>dt:j</em>), collection article (<em>dt:a</em>), book (<em>dt:b</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>doi</strong> </span> <span class="search-explanation" role="cell" role="cell"> Digital Object Identifier (DOI) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ed</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of the editor of a book or special issue </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>en</strong> </span> <span class="search-explanation" role="cell" role="cell"> External document ID: DOI, arXiv ID, ISBN, and others </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>in</strong> </span> <span class="search-explanation" role="cell" role="cell"> zbMATH ID of the corresponding issue </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>la</strong> </span> <span class="search-explanation" role="cell" role="cell"> Language (use name, e.g., <em>la:French</em>, or <a href='https://en.wikipedia.org/wiki/ISO_639-1'>ISO 639-1</a>, e.g., <em>la:FR)</em> </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>li</strong> </span> <span class="search-explanation" role="cell" role="cell"> External link (URL) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>na</strong> </span> <span class="search-explanation" role="cell" role="cell"> Number of authors of the document in question. Interval search with “-” </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>pt</strong> </span> <span class="search-explanation" role="cell" role="cell"> Reviewing state: Reviewed (<em>pt:r</em>), Title Only (<em>pt:t</em>), Pending (<em>pt:p</em>), Scanned Review (<em>pt:s</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>pu</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of the publisher </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>py</strong> </span> <span class="search-explanation" role="cell" role="cell"> Year of publication. Interval search with “-” </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>rft</strong> </span> <span class="search-explanation" role="cell" role="cell"> Text from the references of a document (for phrases use “...”) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>rn</strong> </span> <span class="search-explanation" role="cell" role="cell"> Reviewer ID </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>rv</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name or ID of the reviewer </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>se</strong> </span> <span class="search-explanation" role="cell" role="cell"> Serial ID </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>si</strong> </span> <span class="search-explanation" role="cell" role="cell"> swMATH ID of software referred to in a document </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>so</strong> </span> <span class="search-explanation" role="cell" role="cell"> Bibliographical source, e.g., serial title, volume/issue number, page range, year of publication, ISBN, etc. </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>st</strong> </span> <span class="search-explanation" role="cell" role="cell"> State: is cited (<em>st:c</em>), has references (<em>st:r</em>), has single author (<em>st:s</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>sw</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of software referred to in a document </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ti</strong> </span> <span class="search-explanation" role="cell" role="cell"> Title of the document </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ut</strong> </span> <span class="search-explanation" role="cell" role="cell"> Keywords </span> </div> </div> <p style="margin-bottom:0.5cm;"></p> <h2>Operators</h2> <div id="operators-fields" role="table"> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>a & b</strong></span> <span class="search-explanation" role="cell" role="cell">logical and</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>a | b</strong></span> <span class="search-explanation" role="cell" role="cell">logical or</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>!ab</strong></span> <span class="search-explanation" role="cell" role="cell">logical not</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>abc*</strong></span> <span class="search-explanation" role="cell" role="cell">right wildcard</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>ab c</strong></span> <span class="search-explanation" role="cell" role="cell">phrase</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>(ab c)</strong></span> <span class="search-explanation" role="cell" role="cell">parentheses</span> </div> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/lichtenberg.allan-j" title="Author Profile">Lichtenberg, A. J.</a>; <a href="/authors/lieberman.michael-a" title="Author Profile">Lieberman, M. A.</a></div> <h2 class="title"> <strong>Regular and chaotic dynamics. 2. ed.</strong> <i>(English)</i> <a class="label nowrap" href="/0748.70001">Zbl 0748.70001</a> </h2> <div class="source"> <a href="/serials/3753" title="Series Profile">Applied Mathematical Sciences</a>. 38. New York: Springer-Verlag. xxii, 692 p. (1992). </div> <div class="abstract"><div class="pre">[See the review of the first edition (1983; <a href="/0506.70016">Zbl 0506.70016</a>.)]<br class="zbmathjax-paragraph">The original title of our book, “Regular and stochastic motion”, was chosen to emphasize Hamiltonian dynamics and the physical motion of bodies. The new edition is more evenhanded, with considerably more discussion of dissipative systems and dynamics not involving physical motion. To reflect this partial change of emphasis, we have substituted the more general terms in our title. The common usage of the new terms clarifies the emphasis of the book.<br class="zbmathjax-paragraph">The main change in the book has been to expand the sections on dissipative dynamics, including discussion of renormalization, circle maps, intermittancy, crises, transient chaos, multifractals, reconstruction, and coupled mapping systems. These topics were either mainly in the mathematical literature or essentially unstudied when our first edition was written. The volume of work in these areas has surpassed that in Hamitonian dynamics within the past few years.<br class="zbmathjax-paragraph">We have also made changes in the Hamiltonian sections, adding many new topics such as more general transformation and stability theory, connected stochasticity in two-dimensional maps, converse KAM theory, new topics in diffusion theory, and an approach to equilibrium in many dimensions. Other sections such as mapping models have been revised to take into account new perspectives. We have also corrected a number of misprints and clarified various arguments with the help of colleagues and students. We have again chosen not to treat quantum chaos, partly due to our own lack of acquaintance with the subject. Since writing the first edition, a number of monographs and texts complementary to our book have appeared that are included in the new bibliography.</div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A54000">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A54000">Cited in <strong>374</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A70-02" title="MSC2020">70-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37-02" title="MSC2020">37-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34-02" title="MSC2020">34-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A70Gxx" title="MSC2020">70Gxx</a> </td> <td class="space"> General models, approaches, and methods in mechanics of particles and systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A70Hxx" title="MSC2020">70Hxx</a> </td> <td class="space"> Hamiltonian and Lagrangian mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A70K55" title="MSC2020">70K55</a> </td> <td class="space"> Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34Cxx" title="MSC2020">34Cxx</a> </td> <td class="space"> Qualitative theory for ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37Dxx" title="MSC2020">37Dxx</a> </td> <td class="space"> Dynamical systems with hyperbolic behavior </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37Exx" title="MSC2020">37Exx</a> </td> <td class="space"> Low-dimensional dynamical systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37Hxx" title="MSC2020">37Hxx</a> </td> <td class="space"> Random dynamical systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Arenormalization">renormalization</a>; <a href="/?q=ut%3Acircle+maps">circle maps</a>; <a href="/?q=ut%3Atransient+chaos">transient chaos</a>; <a href="/?q=ut%3Amultifractals">multifractals</a>; <a href="/?q=ut%3Aconverse+KAM+theory">converse KAM theory</a>; <a href="/?q=ut%3Adiffusion+theory">diffusion theory</a>; <a href="/?q=ut%3Aequilibrium+in+many+dimensions">equilibrium in many dimensions</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0506.70016">Zbl 0506.70016</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0748.70001" data-ciurl="/ci/00054000" data-biburl="/bibtex/00054000.bib" data-amsurl="/amsrefs/00054000.bib" data-xmlurl="/xml/00054000.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00054000.pdf" title="Zbl 0748.70001 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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