{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:12:00Z","timestamp":1760242320585,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2017,5,10]],"date-time":"2017-05-10T00:00:00Z","timestamp":1494374400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we apply the differential Galoisian approach to investigate the meromorphic non-integrability of a class of 3D equations in mathematical physics, including Nos\u00e9\u2013Hoover equations, the L\u00fc system, the Rikitake-like system and Rucklidge equations, which are well known in the fields of molecular dynamics, chaotic theory and fluid mechanics, respectively. Our main results show that all these considered systems are, in fact, non-integrable in nearly all parameters.<\/jats:p>","DOI":"10.3390\/e19050211","type":"journal-article","created":{"date-parts":[[2017,5,10]],"date-time":"2017-05-10T12:04:20Z","timestamp":1494417860000},"page":"211","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Meromorphic Non-Integrability of Several 3D Dynamical Systems"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1905-4642","authenticated-orcid":false,"given":"Kaiyin","family":"Huang","sequence":"first","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"},{"name":"State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, China"}]},{"given":"Shaoyun","family":"Shi","sequence":"additional","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"},{"name":"State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, China"}]},{"given":"Wenlei","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"},{"name":"Beijing Computational Science Research Center, Haidian District, Beijing 100094, China"}]}],"member":"1968","published-online":{"date-parts":[[2017,5,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5101","DOI":"10.3390\/e17075101","article-title":"Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu\u2013Morioka System","volume":"17","author":"Leonov","year":"2015","journal-title":"Entropy"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"5561","DOI":"10.3390\/e17085561","article-title":"A New Chaotic System with Positive Topological Entropy","volume":"17","author":"Wang","year":"2015","journal-title":"Entropy"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"7628","DOI":"10.3390\/e17117628","article-title":"A memristor-based complex Lorenz system and its modified projective synchronization","volume":"17","author":"Wang","year":"2015","journal-title":"Entropy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/s002220000066","article-title":"Integrable geodesic flows with positive topological entropy","volume":"140","author":"Bolsinov","year":"2000","journal-title":"Invent. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2003","DOI":"10.1088\/0951-7715\/16\/6\/307","article-title":"Galoisian obstructions to integrability and Melnikov criteria for chaos in two-degree-of-freedom Hamiltonian systems with saddle centres","volume":"16","author":"Yagasaki","year":"2003","journal-title":"Nonlinearity"},{"key":"ref_6","first-page":"259","article-title":"Integrability and nonintegrability of dynamical systems","volume":"122","author":"Goriely","year":"2002","journal-title":"World Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/0378-4371(84)90091-8","article-title":"On the complete and partial integrability of non-Hamiltonian systems","volume":"128","author":"Bountis","year":"1984","journal-title":"Phys. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0370-1573(93)90081-N","article-title":"Painlev\u00e9 analysis, Lie symmetries, and integrability of coupled nonlinear oscillators of polynomial type","volume":"224","author":"Lakshmanan","year":"1993","journal-title":"Phys. Rep."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"63","DOI":"10.2140\/pjm.2007.229.63","article-title":"Multiplicity of invariant algebraic curves in polynomial vector fields","volume":"229","author":"Christopher","year":"2007","journal-title":"Pac. J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/BF01081586","article-title":"Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics I, II","volume":"16","author":"Ziglin","year":"1982","journal-title":"Funct. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Morales-Ruiz, J.J. (1999). Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkh\u00e4user.","DOI":"10.1007\/978-3-0348-0723-4"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1006\/jdeq.1994.1006","article-title":"Picard-Vessiot Theory and Ziglin\u2019s Theorem","volume":"107","year":"1994","journal-title":"J. Differ. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"33","DOI":"10.4310\/MAA.2001.v8.n1.a3","article-title":"Galoisian obstructions to integrability of Hamiltonian systems. I","volume":"8","author":"Ramis","year":"2001","journal-title":"Methods Appl. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"97","DOI":"10.4310\/MAA.2001.v8.n1.a4","article-title":"Galoisian obstructions to integrability of Hamiltonian systems. II","volume":"8","author":"Ramis","year":"2001","journal-title":"Methods Appl. Anal."},{"key":"ref_15","first-page":"5","article-title":"On the infinitesimal geometry of integrable systems","volume":"7","author":"Baider","year":"1996","journal-title":"Mech. Day"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1017\/S0143385700008221","article-title":"Group-theoretic obstructions to integrability","volume":"15","author":"Churchill","year":"1995","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/s10569-013-9514-7","article-title":"Non-integrability of the dumbbell and point mass problem","volume":"117","author":"Maciejewski","year":"2013","journal-title":"Celest. Mech. Dyn. Astronom."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3017","DOI":"10.1016\/j.physleta.2015.09.052","article-title":"Non-integrability of restricted double pendula","volume":"379","author":"Stachowiak","year":"2015","journal-title":"Phys. Lett. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2579","DOI":"10.1088\/0305-4470\/37\/7\/005","article-title":"Non-integrability of the generalized spring-pendulum problem","volume":"37","author":"Maciejewski","year":"2004","journal-title":"J. Phys. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.chaos.2013.04.008","article-title":"Non-integrability of flail triple pendulum","volume":"53","author":"Przybylska","year":"2013","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5518","DOI":"10.1016\/j.jde.2012.01.004","article-title":"Galoisian obstruction to the integrability of general dynamical systems","volume":"252","author":"Li","year":"2012","journal-title":"J. Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1016\/j.jde.2016.10.007","article-title":"Corrigendum to \u201cGaloisian obstruction to the integrability of general dynamical systems\u201d","volume":"262","author":"Li","year":"2017","journal-title":"J. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1323","DOI":"10.1016\/j.crma.2010.10.024","article-title":"Galoisian obstructions to non-Hamiltonian integrability","volume":"348","author":"Ayoul","year":"2010","journal-title":"C. R. Math. Acad. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"022701","DOI":"10.1063\/1.2836412","article-title":"Differential Galois obstructions for integrability of homogeneous Newton equations","volume":"49","author":"Przybylska","year":"2008","journal-title":"J. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/S0375-9601(02)01259-8","article-title":"Non-integrability of ABC flow","volume":"303","author":"Maciejewski","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1023\/A:1026040802018","article-title":"An analytic proof of the nonintegrability of the ABC-flow for A = B = C","volume":"37","author":"Ziglin","year":"2003","journal-title":"Funct. Anal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"616","DOI":"10.1017\/etds.2012.130","article-title":"Meromorphic non-integrability of a steady Stokes flow inside a sphere","volume":"34","author":"Nishiyama","year":"2014","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.1103\/PhysRevA.31.1695","article-title":"Canonical dynamics: Equilibrium phase-space distributions","volume":"31","author":"Hoover","year":"1985","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1007\/s11071-010-9658-x","article-title":"A new chaotic system with fractional order and its projective synchronization","volume":"61","author":"Wu","year":"2010","journal-title":"Nonlinear Dyn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1142\/S0218127402004620","article-title":"A new chaotic attractor coined","volume":"12","author":"Chen","year":"2002","journal-title":"Int. J. Bifurc. Chaos Appl. Sci. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1017\/S0022112092003392","article-title":"Chaos in models of double convection","volume":"237","author":"Rucklidge","year":"1992","journal-title":"J. Fluid Mech."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1007\/s00332-015-9243-z","article-title":"The completely integrable differential systems are essentially linear differential systems","volume":"25","author":"Llibre","year":"2015","journal-title":"J. Nonlinear Sci."},{"key":"ref_33","unstructured":"Van der Put, M., and Singer, M.F. (2012). Galois Theory of Linear Differential Equations, Springer."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0747-7171(86)80010-4","article-title":"An algorithm for solving second order linear homogeneous differential equations","volume":"2","author":"Kovacic","year":"1986","journal-title":"J. Symb. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1941","DOI":"10.1103\/PhysRevA.23.1941","article-title":"Regular-to-irregular transition in conservative Hamiltonian systems: Critical energies and local entropies","volume":"23","author":"Hamilton","year":"1981","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/s00205-006-0029-1","article-title":"Non-ergodicity of the Nos\u00e9\u2013Hoover thermostatted harmonic oscillator","volume":"184","author":"Legoll","year":"2007","journal-title":"Arch. Ration Mech. Anal."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1016\/j.geomphys.2011.02.018","article-title":"Integrability of the Nos\u00e9\u2013Hoover equation","volume":"61","author":"Mahdi","year":"2011","journal-title":"J. Geom. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1250262","DOI":"10.1142\/S0218127412502628","article-title":"Polynomial first integrals for the Chen and L\u00fc systems","volume":"22","author":"Llibre","year":"2012","journal-title":"Int. J. Bifur. Chaos Appl. Sci. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.geomphys.2012.10.003","article-title":"Darboux integrability of the L\u00fc system","volume":"63","author":"Llibre","year":"2013","journal-title":"J. Geom. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1116","DOI":"10.1016\/j.physleta.2009.01.049","article-title":"Degenerate Hopf bifurcations in the L\u00fc system","volume":"373","author":"Mello","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_41","first-page":"1","article-title":"Hopf bifurcations and small amplitude limit cycles in Rucklidge systems","volume":"48","author":"Dias","year":"2013","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s11071-013-1049-7","article-title":"Controlling Rucklidge chaotic system with a single controller using linear feedback and passive control methods","volume":"75","author":"Kocamaz","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1441","DOI":"10.1007\/s11071-014-1389-y","article-title":"Integrability of the Rucklidge system","volume":"77","author":"Lima","year":"2014","journal-title":"Nonlinear Dyn."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/5\/211\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:35:17Z","timestamp":1760207717000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/5\/211"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,10]]},"references-count":43,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2017,5]]}},"alternative-id":["e19050211"],"URL":"https:\/\/doi.org\/10.3390\/e19050211","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,5,10]]}}}