{"id":343,"date":"2021-04-05T07:41:42","date_gmt":"2021-04-05T07:41:42","guid":{"rendered":"https:\/\/vinuni.edu.vn\/research\/?p=343"},"modified":"2022-09-05T07:42:36","modified_gmt":"2022-09-05T07:42:36","slug":"stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials","status":"publish","type":"post","link":"https:\/\/vinuni.edu.vn\/research\/stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials\/","title":{"rendered":"Stability Properties of 1-Dimensional Hamiltonian Lattices with Nonanalytic Potentials"},"content":{"rendered":"<p>Abstract<\/p>\n<p>We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from nonanalytic potentials. In particular, we study the dynamics of a model governed by a \u201cgraphene-type\u201d force law and one inspired by Hollomon\u2019s law describing \u201cwork-hardening\u201d effects in certain elastic materials. Our main aim is to show that, although similarities with the analytic case exist, some of the local and global stability properties of nonanalytic potentials are very different than those encountered in systems with polynomial interactions, as in the case of 1D Fermi\u2013Pasta\u2013Ulam\u2013Tsingou (FPUT) lattices. Our approach is to study the motion in the neighborhood of simple periodic orbits representing continuations of normal modes of the corresponding linear system, as the number of particles\u00a0<span class=\"inline-formula\"><span id=\"MathJax-Element-1-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot; overflow=&quot;scroll&quot; altimg=&quot;eq-00001.gif&quot;&gt;&lt;mi&gt;N&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-1\" class=\"math\"><span id=\"MathJax-Span-2\" class=\"mrow\"><span id=\"MathJax-Span-3\" class=\"mi\">N<\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">N<\/span><\/span><\/span>\u00a0and the total energy\u00a0<span class=\"inline-formula\"><span id=\"MathJax-Element-2-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot; overflow=&quot;scroll&quot; altimg=&quot;eq-00002.gif&quot;&gt;&lt;mi&gt;E&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-4\" class=\"math\"><span id=\"MathJax-Span-5\" class=\"mrow\"><span id=\"MathJax-Span-6\" class=\"mi\">E<\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">E<\/span><\/span><\/span>\u00a0are increased. We find that the graphene-type model is remarkably stable up to escape energy levels where breakdown is expected, while the Hollomon lattice never breaks, yet is unstable at low energies and only attains stability at energies where the harmonic force becomes dominant. We suggest that, since our results hold for large\u00a0<span class=\"inline-formula\"><span id=\"MathJax-Element-3-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot; overflow=&quot;scroll&quot; altimg=&quot;eq-00003.gif&quot;&gt;&lt;mi&gt;N&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-7\" class=\"math\"><span id=\"MathJax-Span-8\" class=\"mrow\"><span id=\"MathJax-Span-9\" class=\"mi\">N<\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">N<\/span><\/span><\/span>, it would be interesting to study analogous phenomena in the continuum limit where 1D lattices become strings.<\/p>\n<p>Authors: Oikonomou, T. and other authors<\/p>\n<p>Read more about the article\u00a0<a href=\"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127420300475\">here<\/a><\/p>\n<p>Read more about the author\u2019s publications\u00a0<a href=\"https:\/\/vinuni.edu.vn\/people\/thomas-oikonomou\/\">here<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Abstract We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from nonanalytic potentials. In particular, we study the dynamics of a model governed by a \u201cgraphene-type\u201d force law and one inspired by Hollomon\u2019s law describing \u201cwork-hardening\u201d effects in certain elastic materials. Our main aim is to [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":344,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[3],"tags":[37],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v19.6.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Stability Properties of 1-Dimensional Hamiltonian Lattices with Nonanalytic Potentials - Vinuni Research Website<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/vinuni.edu.vn\/research\/stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Stability Properties of 1-Dimensional Hamiltonian Lattices with Nonanalytic Potentials - Vinuni Research Website\" \/>\n<meta property=\"og:description\" content=\"Abstract We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from nonanalytic potentials. In particular, we study the dynamics of a model governed by a \u201cgraphene-type\u201d force law and one inspired by Hollomon\u2019s law describing \u201cwork-hardening\u201d effects in certain elastic materials. Our main aim is to [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/vinuni.edu.vn\/research\/stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials\/\" \/>\n<meta property=\"og:site_name\" content=\"Vinuni Research Website\" \/>\n<meta property=\"article:published_time\" content=\"2021-04-05T07:41:42+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-09-05T07:42:36+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/vinuni.edu.vn\/research\/wp-content\/uploads\/2022\/09\/International-Journal-of-Bifurcation-and-Chaos-e1617706722517.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"612\" \/>\n\t<meta property=\"og:image:height\" content=\"380\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"phuong.ntn\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"phuong.ntn\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/vinuni.edu.vn\/research\/stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials\/\",\"url\":\"https:\/\/vinuni.edu.vn\/research\/stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials\/\",\"name\":\"Stability Properties of 1-Dimensional Hamiltonian Lattices with Nonanalytic Potentials - Vinuni Research Website\",\"isPartOf\":{\"@id\":\"https:\/\/vinuni.edu.vn\/research\/#website\"},\"datePublished\":\"2021-04-05T07:41:42+00:00\",\"dateModified\":\"2022-09-05T07:42:36+00:00\",\"author\":{\"@id\":\"https:\/\/vinuni.edu.vn\/research\/#\/schema\/person\/d19200a9f0488a2d6374d8641ee8c085\"},\"breadcrumb\":{\"@id\":\"https:\/\/vinuni.edu.vn\/research\/stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/vinuni.edu.vn\/research\/stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/vinuni.edu.vn\/research\/stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Stability Properties of 1-Dimensional Hamiltonian Lattices with Nonanalytic Potentials\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/vinuni.edu.vn\/research\/#website\",\"url\":\"https:\/\/vinuni.edu.vn\/research\/\",\"name\":\"Vinuni Research Website\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/vinuni.edu.vn\/research\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/vinuni.edu.vn\/research\/#\/schema\/person\/d19200a9f0488a2d6374d8641ee8c085\",\"name\":\"phuong.ntn\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/vinuni.edu.vn\/research\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/678312306781d0830679d2c1a8a112f1?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/678312306781d0830679d2c1a8a112f1?s=96&d=mm&r=g\",\"caption\":\"phuong.ntn\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Stability Properties of 1-Dimensional Hamiltonian Lattices with Nonanalytic Potentials - Vinuni Research Website","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/vinuni.edu.vn\/research\/stability-properties-of-1-dimensional-hamiltonian-lattices-with-nonanalytic-potentials\/","og_locale":"en_US","og_type":"article","og_title":"Stability Properties of 1-Dimensional Hamiltonian Lattices with Nonanalytic Potentials - Vinuni Research Website","og_description":"Abstract We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from nonanalytic potentials. In particular, we study the dynamics of a model governed by a \u201cgraphene-type\u201d force law and one inspired by Hollomon\u2019s law describing \u201cwork-hardening\u201d effects in certain elastic materials. 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