{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:43:02Z","timestamp":1760240582079,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,7,17]],"date-time":"2019-07-17T00:00:00Z","timestamp":1563321600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11671070"],"award-info":[{"award-number":["11671070"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper studies the recognition criterion of the bifurcation problem with trivial solution. The t-equivalence is different from the strong equivalence studied by Golubitsky et al. The difference is that the second component of the differential homeomorphism is not identical. Consider the normal subgroup of t-equivalence group, we obtain the characterization of higher order terms     P ( h )    . In addition, we also explore the properties of intrinsic submodules and the finite determinacy of the bifurcation problem.<\/jats:p>","DOI":"10.3390\/sym11070935","type":"journal-article","created":{"date-parts":[[2019,7,18]],"date-time":"2019-07-18T03:11:42Z","timestamp":1563419502000},"page":"935","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Recognition of the Bifurcation Problem with Trivial Solutions"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8663-6867","authenticated-orcid":false,"given":"Yanqing","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China"},{"name":"School of Science, Hainan Tropical Ocean University, Sanya 572022, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3224-4558","authenticated-orcid":false,"given":"Dejian","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China"},{"name":"School of Science, Hainan Tropical Ocean University, Sanya 572022, China"}]},{"given":"Donghe","family":"Pei","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1038\/246015a0","article-title":"The logic of animal conflict","volume":"246","author":"Smith","year":"1973","journal-title":"Nature"},{"key":"ref_2","first-page":"569","article-title":"The dynamical theory of coevolution: A derivation from stochastic ecological processes","volume":"34","author":"Dieckmann","year":"1997","journal-title":"J. Math. Biol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2024","DOI":"10.1103\/PhysRevLett.78.2024","article-title":"Dynamics of adaptation and evolutionary branching","volume":"78","author":"Geritz","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","first-page":"63","article-title":"A beginner\u2019s guide to adaptive dynamics","volume":"63","author":"Diekmann","year":"2002","journal-title":"Summer Sch. Math. Biol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1111\/j.1420-9101.2005.00948.x","article-title":"20 questions on adaptive dynamics","volume":"18","author":"Waxman","year":"2005","journal-title":"J. Evol. Biol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1007\/s13235-014-0116-0","article-title":"Normal forms and unfoldings of singular strategy functions","volume":"5","author":"Vutha","year":"2015","journal-title":"Dyn. Games Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1007\/s00285-015-0958-0","article-title":"Singularity theory of fitness functions under dimorphism equivalence","volume":"73","author":"Wang","year":"2016","journal-title":"J. Math. Biol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.jtbi.2018.04.002","article-title":"Bifurcation analysis of a mathematical model of atopic dermatitis to determine patient-specific effects of treatments on dynamic phenotypes","volume":"448","author":"Tanaka","year":"2018","journal-title":"J. Theor. Biol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"449","DOI":"10.21042\/AMNS.2017.2.00036","article-title":"Bifurcation Analysis of Hysteretic Systems with Saddle Dynamics","volume":"2","author":"Esteban","year":"2017","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"175","DOI":"10.21042\/AMNS.2018.1.00014","article-title":"Numerical investigation on global dynamics for nonlinear stochastic heat conduction via global random attractors theory","volume":"3","author":"Chen","year":"2018","journal-title":"App. Math. Nonlinear Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/BF01391830","article-title":"Determinacy and unipotency","volume":"88","author":"Bruce","year":"1987","journal-title":"Invent. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1090\/conm\/056\/855086","article-title":"New methods in the classification theory of bifurcation problems","volume":"56","author":"Gaffney","year":"1986","journal-title":"Contemp. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1088\/0951-7715\/1\/1\/009","article-title":"The recognition problem for equivariant singularities","volume":"1","author":"Melbourne","year":"1988","journal-title":"Nonlinearity"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3581","DOI":"10.22436\/jnsa.010.07.18","article-title":"Classification of functions with trivial solutions under t-equivalence","volume":"10","author":"Li","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Golubitsky, M., and Schaeffer, D. (1985). Singularities and Groups in Bifurcation Theory, Springer.","DOI":"10.1007\/978-1-4612-5034-0"},{"key":"ref_16","first-page":"24","article-title":"Global branchings in power systems","volume":"6","author":"Yuan","year":"1994","journal-title":"Automat. Electron. Power Syst."},{"key":"ref_17","first-page":"42","article-title":"On Bifurcation of a Class of Nonlinear Differential System","volume":"19","author":"Xiong","year":"2007","journal-title":"J. North China Univ. Technol."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/7\/935\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:06:44Z","timestamp":1760188004000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/7\/935"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,17]]},"references-count":17,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2019,7]]}},"alternative-id":["sym11070935"],"URL":"https:\/\/doi.org\/10.3390\/sym11070935","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,7,17]]}}}