Cyclicity of slow–fast cycles with one self-intersection point and two nilpotent contact points* Article Swipe
Jinhui Yao
,
Jicai Huang
,
Renato Huzak
,
Shigui Ruan
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.1088/1361-6544/ad7c11
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.1088/1361-6544/ad7c11
In this paper, we study the cyclicity of slow–fast cycles with one self-intersection point and two nilpotent contact points in planar slow–fast systems, where the nilpotent contact point is a jump point or a slow–fast Hopf point. These slow–fast cycles can be classified into three cases based on the two nilpotent contact points: (i) both are generic jump points, (ii) one is a generic jump point and the other is a slow–fast Hopf point, and (iii) both are slow–fast Hopf points. By using slow divergence integrals and entry–exit functions, we show that the cyclicity of slow–fast cycles with one self-intersection point and two generic jump points (or one generic jump point and one slow–fast Hopf point) is at most two, and the cyclicity of slow–fast cycles with one self-intersection point and two slow–fast Hopf points is at most three under some specific conditions. Finally, we apply the main results to two predator-prey models.
Related Topics
Concepts
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1088/1361-6544/ad7c11
- OA Status
- green
- Cited By
- 2
- References
- 32
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4402887287
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4402887287Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1088/1361-6544/ad7c11Digital Object Identifier
- Title
-
Cyclicity of slow–fast cycles with one self-intersection point and two nilpotent contact points*Work title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2024Year of publication
- Publication date
-
2024-09-26Full publication date if available
- Authors
-
Jinhui Yao, Jicai Huang, Renato Huzak, Shigui RuanList of authors in order
- Landing page
-
https://doi.org/10.1088/1361-6544/ad7c11Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://hdl.handle.net/1942/44406Direct OA link when available
- Concepts
-
Mathematics, Nilpotent, Intersection (aeronautics), Combinatorics, Point (geometry), Pure mathematics, Geometry, Geography, CartographyTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
2Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 2Per-year citation counts (last 5 years)
- References (count)
-
32Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4402887287 |
|---|---|
| doi | https://doi.org/10.1088/1361-6544/ad7c11 |
| ids.doi | https://doi.org/10.1088/1361-6544/ad7c11 |
| ids.openalex | https://openalex.org/W4402887287 |
| fwci | 1.08467083 |
| type | article |
| title | Cyclicity of slow–fast cycles with one self-intersection point and two nilpotent contact points* |
| biblio.issue | 11 |
| biblio.volume | 37 |
| biblio.last_page | 115007 |
| biblio.first_page | 115007 |
| grants[0].funder | https://openalex.org/F4320321001 |
| grants[0].award_id | No. 11871235 |
| grants[0].funder_display_name | National Natural Science Foundation of China |
| topics[0].id | https://openalex.org/T11520 |
| topics[0].field.id | https://openalex.org/fields/31 |
| topics[0].field.display_name | Physics and Astronomy |
| topics[0].score | 0.9965999722480774 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/3109 |
| topics[0].subfield.display_name | Statistical and Nonlinear Physics |
| topics[0].display_name | Advanced Thermodynamics and Statistical Mechanics |
| topics[1].id | https://openalex.org/T10492 |
| topics[1].field.id | https://openalex.org/fields/13 |
| topics[1].field.display_name | Biochemistry, Genetics and Molecular Biology |
| topics[1].score | 0.9890000224113464 |
| topics[1].domain.id | https://openalex.org/domains/1 |
| topics[1].domain.display_name | Life Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/1307 |
| topics[1].subfield.display_name | Cell Biology |
| topics[1].display_name | Microtubule and mitosis dynamics |
| topics[2].id | https://openalex.org/T12396 |
| topics[2].field.id | https://openalex.org/fields/26 |
| topics[2].field.display_name | Mathematics |
| topics[2].score | 0.9861000180244446 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/2608 |
| topics[2].subfield.display_name | Geometry and Topology |
| topics[2].display_name | Advanced Differential Equations and Dynamical Systems |
| funders[0].id | https://openalex.org/F4320321001 |
| funders[0].ror | https://ror.org/01h0zpd94 |
| funders[0].display_name | National Natural Science Foundation of China |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C33923547 |
| concepts[0].level | 0 |
| concepts[0].score | 0.8784730434417725 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[0].display_name | Mathematics |
| concepts[1].id | https://openalex.org/C50555996 |
| concepts[1].level | 2 |
| concepts[1].score | 0.8517273664474487 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q840023 |
| concepts[1].display_name | Nilpotent |
| concepts[2].id | https://openalex.org/C64543145 |
| concepts[2].level | 2 |
| concepts[2].score | 0.6751993298530579 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q162942 |
| concepts[2].display_name | Intersection (aeronautics) |
| concepts[3].id | https://openalex.org/C114614502 |
| concepts[3].level | 1 |
| concepts[3].score | 0.5054244995117188 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q76592 |
| concepts[3].display_name | Combinatorics |
| concepts[4].id | https://openalex.org/C28719098 |
| concepts[4].level | 2 |
| concepts[4].score | 0.47029682993888855 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q44946 |
| concepts[4].display_name | Point (geometry) |
| concepts[5].id | https://openalex.org/C202444582 |
| concepts[5].level | 1 |
| concepts[5].score | 0.4415380358695984 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q837863 |
| concepts[5].display_name | Pure mathematics |
| concepts[6].id | https://openalex.org/C2524010 |
| concepts[6].level | 1 |
| concepts[6].score | 0.2799951434135437 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q8087 |
| concepts[6].display_name | Geometry |
| concepts[7].id | https://openalex.org/C205649164 |
| concepts[7].level | 0 |
| concepts[7].score | 0.054463326930999756 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q1071 |
| concepts[7].display_name | Geography |
| concepts[8].id | https://openalex.org/C58640448 |
| concepts[8].level | 1 |
| concepts[8].score | 0.0 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q42515 |
| concepts[8].display_name | Cartography |
| keywords[0].id | https://openalex.org/keywords/mathematics |
| keywords[0].score | 0.8784730434417725 |
| keywords[0].display_name | Mathematics |
| keywords[1].id | https://openalex.org/keywords/nilpotent |
| keywords[1].score | 0.8517273664474487 |
| keywords[1].display_name | Nilpotent |
| keywords[2].id | https://openalex.org/keywords/intersection |
| keywords[2].score | 0.6751993298530579 |
| keywords[2].display_name | Intersection (aeronautics) |
| keywords[3].id | https://openalex.org/keywords/combinatorics |
| keywords[3].score | 0.5054244995117188 |
| keywords[3].display_name | Combinatorics |
| keywords[4].id | https://openalex.org/keywords/point |
| keywords[4].score | 0.47029682993888855 |
| keywords[4].display_name | Point (geometry) |
| keywords[5].id | https://openalex.org/keywords/pure-mathematics |
| keywords[5].score | 0.4415380358695984 |
| keywords[5].display_name | Pure mathematics |
| keywords[6].id | https://openalex.org/keywords/geometry |
| keywords[6].score | 0.2799951434135437 |
| keywords[6].display_name | Geometry |
| keywords[7].id | https://openalex.org/keywords/geography |
| keywords[7].score | 0.054463326930999756 |
| keywords[7].display_name | Geography |
| language | en |
| locations[0].id | doi:10.1088/1361-6544/ad7c11 |
| locations[0].is_oa | False |
| locations[0].source.id | https://openalex.org/S158842918 |
| locations[0].source.issn | 0951-7715, 1361-6544 |
| locations[0].source.type | journal |
| locations[0].source.is_oa | False |
| locations[0].source.issn_l | 0951-7715 |
| locations[0].source.is_core | True |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | Nonlinearity |
| locations[0].source.host_organization | https://openalex.org/P4310320083 |
| locations[0].source.host_organization_name | IOP Publishing |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310320083 |
| locations[0].license | |
| locations[0].pdf_url | |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Nonlinearity |
| locations[0].landing_page_url | https://doi.org/10.1088/1361-6544/ad7c11 |
| locations[1].id | pmh:oai:documentserver.uhasselt.be:1942/44406 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306401926 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | False |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | Document Server@UHasselt (UHasselt) |
| locations[1].source.host_organization | https://openalex.org/I878454856 |
| locations[1].source.host_organization_name | Hasselt University |
| locations[1].source.host_organization_lineage | https://openalex.org/I878454856 |
| locations[1].license | other-oa |
| locations[1].pdf_url | |
| locations[1].version | submittedVersion |
| locations[1].raw_type | info:eu-repo/semantics/article |
| locations[1].license_id | https://openalex.org/licenses/other-oa |
| locations[1].is_accepted | False |
| locations[1].is_published | False |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | http://hdl.handle.net/1942/44406 |
| indexed_in | crossref |
| authorships[0].author.id | https://openalex.org/A5062391972 |
| authorships[0].author.orcid | https://orcid.org/0000-0002-4747-1905 |
| authorships[0].author.display_name | Jinhui Yao |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Jinhui Yao |
| authorships[0].is_corresponding | False |
| authorships[1].author.id | https://openalex.org/A5054826576 |
| authorships[1].author.orcid | https://orcid.org/0000-0002-8910-5233 |
| authorships[1].author.display_name | Jicai Huang |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Jicai Huang |
| authorships[1].is_corresponding | False |
| authorships[2].author.id | https://openalex.org/A5080769869 |
| authorships[2].author.orcid | https://orcid.org/0000-0002-3391-7777 |
| authorships[2].author.display_name | Renato Huzak |
| authorships[2].author_position | middle |
| authorships[2].raw_author_name | Renato Huzak |
| authorships[2].is_corresponding | False |
| authorships[3].author.id | https://openalex.org/A5091820514 |
| authorships[3].author.orcid | https://orcid.org/0000-0002-6348-8205 |
| authorships[3].author.display_name | Shigui Ruan |
| authorships[3].author_position | last |
| authorships[3].raw_author_name | Shigui Ruan |
| authorships[3].is_corresponding | False |
| has_content.pdf | False |
| has_content.grobid_xml | False |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | http://hdl.handle.net/1942/44406 |
| open_access.oa_status | green |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Cyclicity of slow–fast cycles with one self-intersection point and two nilpotent contact points* |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T11520 |
| primary_topic.field.id | https://openalex.org/fields/31 |
| primary_topic.field.display_name | Physics and Astronomy |
| primary_topic.score | 0.9965999722480774 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/3109 |
| primary_topic.subfield.display_name | Statistical and Nonlinear Physics |
| primary_topic.display_name | Advanced Thermodynamics and Statistical Mechanics |
| related_works | https://openalex.org/W4391375266, https://openalex.org/W2348909947, https://openalex.org/W4292672442, https://openalex.org/W1977169298, https://openalex.org/W2371953206, https://openalex.org/W2382500641, https://openalex.org/W4230797523, https://openalex.org/W2801497733, https://openalex.org/W1789793208, https://openalex.org/W2291386138 |
| cited_by_count | 2 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 2 |
| locations_count | 2 |
| best_oa_location.id | pmh:oai:documentserver.uhasselt.be:1942/44406 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4306401926 |
| best_oa_location.source.issn | |
| best_oa_location.source.type | repository |
| best_oa_location.source.is_oa | False |
| best_oa_location.source.issn_l | |
| best_oa_location.source.is_core | False |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | Document Server@UHasselt (UHasselt) |
| best_oa_location.source.host_organization | https://openalex.org/I878454856 |
| best_oa_location.source.host_organization_name | Hasselt University |
| best_oa_location.source.host_organization_lineage | https://openalex.org/I878454856 |
| best_oa_location.license | other-oa |
| best_oa_location.pdf_url | |
| best_oa_location.version | submittedVersion |
| best_oa_location.raw_type | info:eu-repo/semantics/article |
| best_oa_location.license_id | https://openalex.org/licenses/other-oa |
| best_oa_location.is_accepted | False |
| best_oa_location.is_published | False |
| best_oa_location.raw_source_name | |
| best_oa_location.landing_page_url | http://hdl.handle.net/1942/44406 |
| primary_location.id | doi:10.1088/1361-6544/ad7c11 |
| primary_location.is_oa | False |
| primary_location.source.id | https://openalex.org/S158842918 |
| primary_location.source.issn | 0951-7715, 1361-6544 |
| primary_location.source.type | journal |
| primary_location.source.is_oa | False |
| primary_location.source.issn_l | 0951-7715 |
| primary_location.source.is_core | True |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | Nonlinearity |
| primary_location.source.host_organization | https://openalex.org/P4310320083 |
| primary_location.source.host_organization_name | IOP Publishing |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310320083 |
| primary_location.license | |
| primary_location.pdf_url | |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Nonlinearity |
| primary_location.landing_page_url | https://doi.org/10.1088/1361-6544/ad7c11 |
| publication_date | 2024-09-26 |
| publication_year | 2024 |
| referenced_works | https://openalex.org/W2043607789, https://openalex.org/W2808199969, https://openalex.org/W2073062893, https://openalex.org/W1971039616, https://openalex.org/W1836029049, https://openalex.org/W2130344039, https://openalex.org/W2963769644, https://openalex.org/W2062272038, https://openalex.org/W2033338472, https://openalex.org/W2018185874, https://openalex.org/W1965868683, https://openalex.org/W2336126067, https://openalex.org/W3176541892, https://openalex.org/W2293963809, https://openalex.org/W2898492458, https://openalex.org/W4220739604, https://openalex.org/W1986739294, https://openalex.org/W1981961598, https://openalex.org/W2089995439, https://openalex.org/W2010265353, https://openalex.org/W2069773303, https://openalex.org/W1534612556, https://openalex.org/W2100990586, https://openalex.org/W2793355291, https://openalex.org/W2944012157, https://openalex.org/W2334229214, https://openalex.org/W6884311957, https://openalex.org/W4396686555, https://openalex.org/W4312222532, https://openalex.org/W3145000948, https://openalex.org/W4377148681, https://openalex.org/W3103319120 |
| referenced_works_count | 32 |
| abstract_inverted_index.a | 30, 34, 63, 71 |
| abstract_inverted_index.By | 82 |
| abstract_inverted_index.In | 1 |
| abstract_inverted_index.at | 118, 137 |
| abstract_inverted_index.be | 42 |
| abstract_inverted_index.in | 20 |
| abstract_inverted_index.is | 29, 62, 70, 117, 136 |
| abstract_inverted_index.of | 8, 95, 124 |
| abstract_inverted_index.on | 48 |
| abstract_inverted_index.or | 33 |
| abstract_inverted_index.to | 150 |
| abstract_inverted_index.we | 4, 90, 145 |
| abstract_inverted_index.(i) | 54 |
| abstract_inverted_index.(or | 107 |
| abstract_inverted_index.and | 15, 67, 75, 87, 102, 112, 121, 131 |
| abstract_inverted_index.are | 56, 78 |
| abstract_inverted_index.can | 41 |
| abstract_inverted_index.one | 12, 61, 99, 108, 113, 128 |
| abstract_inverted_index.the | 6, 25, 49, 68, 93, 122, 147 |
| abstract_inverted_index.two | 16, 50, 103, 132, 151 |
| abstract_inverted_index.(ii) | 60 |
| abstract_inverted_index.Hopf | 36, 73, 80, 115, 134 |
| abstract_inverted_index.both | 55, 77 |
| abstract_inverted_index.into | 44 |
| abstract_inverted_index.jump | 31, 58, 65, 105, 110 |
| abstract_inverted_index.main | 148 |
| abstract_inverted_index.most | 119, 138 |
| abstract_inverted_index.show | 91 |
| abstract_inverted_index.slow | 84 |
| abstract_inverted_index.some | 141 |
| abstract_inverted_index.that | 92 |
| abstract_inverted_index.this | 2 |
| abstract_inverted_index.two, | 120 |
| abstract_inverted_index.with | 11, 98, 127 |
| abstract_inverted_index.(iii) | 76 |
| abstract_inverted_index.These | 38 |
| abstract_inverted_index.apply | 146 |
| abstract_inverted_index.based | 47 |
| abstract_inverted_index.cases | 46 |
| abstract_inverted_index.other | 69 |
| abstract_inverted_index.point | 14, 28, 32, 66, 101, 111, 130 |
| abstract_inverted_index.study | 5 |
| abstract_inverted_index.three | 45, 139 |
| abstract_inverted_index.under | 140 |
| abstract_inverted_index.using | 83 |
| abstract_inverted_index.where | 24 |
| abstract_inverted_index.cycles | 10, 40, 97, 126 |
| abstract_inverted_index.paper, | 3 |
| abstract_inverted_index.planar | 21 |
| abstract_inverted_index.point) | 116 |
| abstract_inverted_index.point, | 74 |
| abstract_inverted_index.point. | 37 |
| abstract_inverted_index.points | 19, 106, 135 |
| abstract_inverted_index.contact | 18, 27, 52 |
| abstract_inverted_index.generic | 57, 64, 104, 109 |
| abstract_inverted_index.models. | 153 |
| abstract_inverted_index.points, | 59 |
| abstract_inverted_index.points. | 81 |
| abstract_inverted_index.points: | 53 |
| abstract_inverted_index.results | 149 |
| abstract_inverted_index.Abstract | 0 |
| abstract_inverted_index.Finally, | 144 |
| abstract_inverted_index.specific | 142 |
| abstract_inverted_index.systems, | 23 |
| abstract_inverted_index.cyclicity | 7, 94, 123 |
| abstract_inverted_index.integrals | 86 |
| abstract_inverted_index.nilpotent | 17, 26, 51 |
| abstract_inverted_index.classified | 43 |
| abstract_inverted_index.divergence | 85 |
| abstract_inverted_index.functions, | 89 |
| abstract_inverted_index.conditions. | 143 |
| abstract_inverted_index.slow–fast | 9, 22, 35, 39, 72, 79, 96, 114, 125, 133 |
| abstract_inverted_index.entry–exit | 88 |
| abstract_inverted_index.predator-prey | 152 |
| abstract_inverted_index.self-intersection | 13, 100, 129 |
| cited_by_percentile_year.max | 97 |
| cited_by_percentile_year.min | 95 |
| countries_distinct_count | 0 |
| institutions_distinct_count | 4 |
| citation_normalized_percentile.value | 0.7264135 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |