{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T18:00:11Z","timestamp":1790704811550,"version":"4.1.0"},"reference-count":21,"publisher":"Public Library of Science (PLoS)","issue":"9","license":[{"start":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T00:00:00Z","timestamp":1790640000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"<jats:p>This paper addresses the problem of Lyapunov stability for equilibrium solutions in periodic Hamiltonian systems under degenerate conditions. The degeneracy occurs in resonant cases when the leading coefficient function in the normal form possesses only zeros of high multiplicity, rendering standard criteria inapplicable. By employing Lie normal form theory, we establish two general stability criteria that resolve the stability problem for nearly all such degenerate scenarios. Our results explicitly demonstrate the decisive role of higher-order terms and provide easily verifiable conditions that generalize several known results in the literature.<\/jats:p>","DOI":"10.1371\/journal.pone.0359575","type":"journal-article","created":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T17:34:08Z","timestamp":1790703248000},"page":"e0359575","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the stability of equilibria in degenerate periodic Hamiltonian systems"],"prefix":"10.1371","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9316-030X","authenticated-orcid":true,"given":"Nina","family":"Xue","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Weifang University","place":["Weifang, Shandong, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2026,9,29]]},"reference":[{"key":"pone.0359575.ref001","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s10884-010-9176-z","article-title":"Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems with n-degrees of freedom in the case of single resonance","volume":"22","author":"F dos Santos","year":"2010","journal-title":"J Dyn Differ Equ"},{"key":"pone.0359575.ref002","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1070\/RD2005v010n01ABEH000303","article-title":"Stability of equilibrium solutions of periodic Hamiltonian systems under third and fourth order resonances","volume":"10","author":"C Vidal","year":"2005","journal-title":"Regul Chaotic Dyn"},{"key":"pone.0359575.ref003","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1016\/j.jde.2017.09.018","article-title":"The linearization of periodic Hamiltonian systems with one degree of freedom under the Diophantine condition","volume":"264","author":"N Xue","year":"2018","journal-title":"J Differ Equ"},{"key":"pone.0359575.ref004","first-page":"6260253","article-title":"On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schr\u00f6dinger Equation","author":"N Xue","year":"2020","journal-title":"J Funct Spaces"},{"key":"pone.0359575.ref005","first-page":"247","article-title":"The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case","volume":"2","author":"VI Arnold","year":"1961","journal-title":"Sov Math Dokl"},{"key":"pone.0359575.ref006","unstructured":"Moser J. On invariant curves of area preserving mappings of an annulus. Nachr Akad Wiss G\u00f6ttingen Math Phys Kl. 1962;:1\u201320."},{"key":"pone.0359575.ref007","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1007\/BF01048263","article-title":"The twist coefficient of periodic solutions of a time-dependent Newton\u2019s equation","volume":"4","author":"R Ortega","year":"1992","journal-title":"J Dyn Differ Equ"},{"issue":"5","key":"pone.0359575.ref008","doi-asserted-by":"crossref","first-page":"1393","DOI":"10.1137\/S003614109223920X","article-title":"The Stability of the Equilibrium of a Nonlinear Hill\u2019s Equation","volume":"25","author":"R Ortega","year":"1994","journal-title":"SIAM J Math Anal"},{"key":"pone.0359575.ref009","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1006\/jdeq.1996.0103","article-title":"Periodic solutions of a Newtonian equation: stability by the third approximation","volume":"128","author":"R Ortega","year":"1996","journal-title":"J Differ Equ"},{"key":"pone.0359575.ref010","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/B978-044451861-3\/50008-2","article-title":"The stability of the equilibrium: A search for the right approximation","volume-title":"10 Mathematical Essays on Approximation in Analysis and Topology","author":"R Ortega","year":"2005"},{"issue":"4","key":"pone.0359575.ref011","doi-asserted-by":"crossref","first-page":"844","DOI":"10.1137\/S003614100241037X","article-title":"Twist Character of the Least Amplitude Periodic Solution of the Forced Pendulum","volume":"35","author":"J Lei","year":"2003","journal-title":"SIAM J Math Anal"},{"key":"pone.0359575.ref012","doi-asserted-by":"crossref","first-page":"890","DOI":"10.1007\/PL00001528","article-title":"Parabolic fixed points and stability criteria for nonlinear Hill\u2019s equation","volume":"51","author":"D N\u00fa\u00f1ez","year":"2000","journal-title":"Z Angew Math Phys"},{"issue":"3","key":"pone.0359575.ref013","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1515\/ans-2002-0305","article-title":"Twist Solutions of a Hill\u2019s Equation with Singular Term","volume":"2","author":"PJ Torres","year":"2002","journal-title":"Advanced Nonlinear Studies"},{"key":"pone.0359575.ref014","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1017\/S0308210505000739","article-title":"Existence and stability of periodic solutions for second-order semilinear differential equations with a singular nonlinearity","volume":"137","author":"PJ Torres","year":"2007","journal-title":"Proc R Soc Edinb Sect A"},{"issue":"6","key":"pone.0359575.ref015","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1134\/S1560354715060015","article-title":"On the stability of periodic Hamiltonian systems with one degree of freedom in the case of degeneracy","volume":"20","author":"BS Bardin","year":"2015","journal-title":"Regul Chaot Dyn"},{"issue":"5","key":"pone.0359575.ref016","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1088\/0951-7715\/12\/5\/309","article-title":"Stability of equilibria and fixed points of conservative systems","volume":"12","author":"HE Cabral","year":"1999","journal-title":"Nonlinearity"},{"key":"pone.0359575.ref017","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/s10569-013-9488-5","article-title":"Stability of equilibrium solutions in the critical case of even-order resonance in periodic Hamiltonian systems with one degree of freedom","volume":"116","author":"JE Mansilla","year":"2013","journal-title":"Celest Mech Dyn Astron"},{"key":"pone.0359575.ref018","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1016\/S0021-8928(97)00045-2","article-title":"The critical case of fourth-order resonance in a Hamiltonian system with one degree of freedom","volume":"61","author":"AP Markeyev","year":"1997","journal-title":"J Appl Math Mech"},{"key":"pone.0359575.ref019","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1016\/S0021-8928(01)00068-5","article-title":"The problem of the stability of the equilibrium position of a Hamiltonian system at 3:1 resonance","volume":"65","author":"AP Markeyev","year":"2001","journal-title":"J Appl Math Mech"},{"key":"pone.0359575.ref020","doi-asserted-by":"crossref","first-page":"3880","DOI":"10.1016\/j.jde.2015.01.044","article-title":"Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems in the case of multiple resonances","volume":"258","author":"F dos Santos","year":"2015","journal-title":"J Differ Equ"},{"key":"pone.0359575.ref021","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-4073-8","volume-title":"Introduction to Hamiltonian dynamical systems and the n-body problem","author":"KR Meyer","year":"1992"}],"container-title":["PLOS One"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0359575","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T17:34:11Z","timestamp":1790703251000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0359575"}},"subtitle":[],"editor":[{"given":"Zeheng","family":"Wang","sequence":"first","affiliation":[{"name":"Commonwealth Scientific and Industrial Research Organisation","place":["AUSTRALIA"]}],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2026,9,29]]},"references-count":21,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9,29]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pone.0359575","relation":{},"ISSN":["1932-6203"],"issn-type":[{"value":"1932-6203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,29]]}}}