Synchronization and Random Attractors in Reaction Jump Processes Article Swipe
Maximilian Engel
,
Guillermo Olicón-Méndez
,
Nathalie Wehlitz
,
Stefanie Winkelmann
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.1007/s10884-023-10345-4
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.1007/s10884-023-10345-4
This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. Based on Gillespie’s stochastic simulation algorithm, a corresponding random dynamical system is formulated in a two-step procedure, at first for the states of the embedded discrete-time Markov chain and then for the augmented Markov chain including random jump times. We uncover a time-shifted synchronization in the sense that—after some initial waiting time—one trajectory exactly replicates another one with a certain time delay. Whether or not such a synchronization behavior occurs depends on the combination of the initial states. We prove this partial time-shifted synchronization for the special setting of a birth-death process by analyzing the corresponding two-point motion of the embedded Markov chain and determine the structure of the associated random attractor. In this context, we also provide general results on existence and form of random attractors for discrete-time, discrete-space random dynamical systems.
Related Topics
Concepts
Mathematics
Markov chain
Attractor
Synchronization (alternating current)
Ordinary differential equation
Context (archaeology)
Markov process
Time reversibility
Statistical physics
Discrete time and continuous time
Jump
Jump process
Applied mathematics
Variable-order Markov model
Markov model
Mathematical analysis
Topology (electrical circuits)
Differential equation
Combinatorics
Statistics
Physics
Paleontology
Quantum mechanics
Biology
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1007/s10884-023-10345-4
- OA Status
- hybrid
- Cited By
- 2
- References
- 39
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4392123601
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4392123601Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1007/s10884-023-10345-4Digital Object Identifier
- Title
-
Synchronization and Random Attractors in Reaction Jump ProcessesWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2024Year of publication
- Publication date
-
2024-02-23Full publication date if available
- Authors
-
Maximilian Engel, Guillermo Olicón-Méndez, Nathalie Wehlitz, Stefanie WinkelmannList of authors in order
- Landing page
-
https://doi.org/10.1007/s10884-023-10345-4Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
hybridOpen access status per OpenAlex
- OA URL
-
https://doi.org/10.1007/s10884-023-10345-4Direct OA link when available
- Concepts
-
Mathematics, Markov chain, Attractor, Synchronization (alternating current), Ordinary differential equation, Context (archaeology), Markov process, Time reversibility, Statistical physics, Discrete time and continuous time, Jump, Jump process, Applied mathematics, Variable-order Markov model, Markov model, Mathematical analysis, Topology (electrical circuits), Differential equation, Combinatorics, Statistics, Physics, Paleontology, Quantum mechanics, BiologyTop 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)
-
39Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4392123601 |
|---|---|
| doi | https://doi.org/10.1007/s10884-023-10345-4 |
| ids.doi | https://doi.org/10.1007/s10884-023-10345-4 |
| ids.openalex | https://openalex.org/W4392123601 |
| fwci | 0.96049445 |
| type | article |
| title | Synchronization and Random Attractors in Reaction Jump Processes |
| awards[0].id | https://openalex.org/G3838484177 |
| awards[0].funder_id | https://openalex.org/F4320320879 |
| awards[0].display_name | |
| awards[0].funder_award_id | EXC-2046/1, project 390685689 |
| awards[0].funder_display_name | Deutsche Forschungsgemeinschaft |
| biblio.issue | 2 |
| biblio.volume | 37 |
| biblio.last_page | 1242 |
| biblio.first_page | 1207 |
| topics[0].id | https://openalex.org/T10621 |
| topics[0].field.id | https://openalex.org/fields/13 |
| topics[0].field.display_name | Biochemistry, Genetics and Molecular Biology |
| topics[0].score | 0.9998000264167786 |
| topics[0].domain.id | https://openalex.org/domains/1 |
| topics[0].domain.display_name | Life Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/1312 |
| topics[0].subfield.display_name | Molecular Biology |
| topics[0].display_name | Gene Regulatory Network Analysis |
| topics[1].id | https://openalex.org/T11187 |
| topics[1].field.id | https://openalex.org/fields/17 |
| topics[1].field.display_name | Computer Science |
| topics[1].score | 0.9954000115394592 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/1705 |
| topics[1].subfield.display_name | Computer Networks and Communications |
| topics[1].display_name | Nonlinear Dynamics and Pattern Formation |
| topics[2].id | https://openalex.org/T11513 |
| topics[2].field.id | https://openalex.org/fields/31 |
| topics[2].field.display_name | Physics and Astronomy |
| topics[2].score | 0.967199981212616 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/3109 |
| topics[2].subfield.display_name | Statistical and Nonlinear Physics |
| topics[2].display_name | stochastic dynamics and bifurcation |
| funders[0].id | https://openalex.org/F4320320879 |
| funders[0].ror | https://ror.org/018mejw64 |
| funders[0].display_name | Deutsche Forschungsgemeinschaft |
| funders[1].id | https://openalex.org/F4320324521 |
| funders[1].ror | |
| funders[1].display_name | Freie Universität Berlin |
| is_xpac | False |
| apc_list.value | 2290 |
| apc_list.currency | EUR |
| apc_list.value_usd | 2890 |
| apc_paid.value | 2290 |
| apc_paid.currency | EUR |
| apc_paid.value_usd | 2890 |
| concepts[0].id | https://openalex.org/C33923547 |
| concepts[0].level | 0 |
| concepts[0].score | 0.6396316289901733 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[0].display_name | Mathematics |
| concepts[1].id | https://openalex.org/C98763669 |
| concepts[1].level | 2 |
| concepts[1].score | 0.6379321813583374 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q176645 |
| concepts[1].display_name | Markov chain |
| concepts[2].id | https://openalex.org/C164380108 |
| concepts[2].level | 2 |
| concepts[2].score | 0.6033686995506287 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q507187 |
| concepts[2].display_name | Attractor |
| concepts[3].id | https://openalex.org/C2778562939 |
| concepts[3].level | 3 |
| concepts[3].score | 0.5447115302085876 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q1298791 |
| concepts[3].display_name | Synchronization (alternating current) |
| concepts[4].id | https://openalex.org/C51544822 |
| concepts[4].level | 3 |
| concepts[4].score | 0.5237728953361511 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q465274 |
| concepts[4].display_name | Ordinary differential equation |
| concepts[5].id | https://openalex.org/C2779343474 |
| concepts[5].level | 2 |
| concepts[5].score | 0.5026881694793701 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q3109175 |
| concepts[5].display_name | Context (archaeology) |
| concepts[6].id | https://openalex.org/C159886148 |
| concepts[6].level | 2 |
| concepts[6].score | 0.49864912033081055 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q176645 |
| concepts[6].display_name | Markov process |
| concepts[7].id | https://openalex.org/C98735484 |
| concepts[7].level | 5 |
| concepts[7].score | 0.47013670206069946 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q186971 |
| concepts[7].display_name | Time reversibility |
| concepts[8].id | https://openalex.org/C121864883 |
| concepts[8].level | 1 |
| concepts[8].score | 0.4443860352039337 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q677916 |
| concepts[8].display_name | Statistical physics |
| concepts[9].id | https://openalex.org/C55689738 |
| concepts[9].level | 2 |
| concepts[9].score | 0.441024512052536 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q15963867 |
| concepts[9].display_name | Discrete time and continuous time |
| concepts[10].id | https://openalex.org/C2780695682 |
| concepts[10].level | 2 |
| concepts[10].score | 0.4321504533290863 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q4005959 |
| concepts[10].display_name | Jump |
| concepts[11].id | https://openalex.org/C2776825372 |
| concepts[11].level | 3 |
| concepts[11].score | 0.42752930521965027 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q3922411 |
| concepts[11].display_name | Jump process |
| concepts[12].id | https://openalex.org/C28826006 |
| concepts[12].level | 1 |
| concepts[12].score | 0.3547423481941223 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q33521 |
| concepts[12].display_name | Applied mathematics |
| concepts[13].id | https://openalex.org/C54907487 |
| concepts[13].level | 4 |
| concepts[13].score | 0.3419708013534546 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q7915688 |
| concepts[13].display_name | Variable-order Markov model |
| concepts[14].id | https://openalex.org/C163836022 |
| concepts[14].level | 3 |
| concepts[14].score | 0.26474133133888245 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q6771326 |
| concepts[14].display_name | Markov model |
| concepts[15].id | https://openalex.org/C134306372 |
| concepts[15].level | 1 |
| concepts[15].score | 0.18507987260818481 |
| concepts[15].wikidata | https://www.wikidata.org/wiki/Q7754 |
| concepts[15].display_name | Mathematical analysis |
| concepts[16].id | https://openalex.org/C184720557 |
| concepts[16].level | 2 |
| concepts[16].score | 0.16976135969161987 |
| concepts[16].wikidata | https://www.wikidata.org/wiki/Q7825049 |
| concepts[16].display_name | Topology (electrical circuits) |
| concepts[17].id | https://openalex.org/C78045399 |
| concepts[17].level | 2 |
| concepts[17].score | 0.12430712580680847 |
| concepts[17].wikidata | https://www.wikidata.org/wiki/Q11214 |
| concepts[17].display_name | Differential equation |
| concepts[18].id | https://openalex.org/C114614502 |
| concepts[18].level | 1 |
| concepts[18].score | 0.1205810010433197 |
| concepts[18].wikidata | https://www.wikidata.org/wiki/Q76592 |
| concepts[18].display_name | Combinatorics |
| concepts[19].id | https://openalex.org/C105795698 |
| concepts[19].level | 1 |
| concepts[19].score | 0.07646229863166809 |
| concepts[19].wikidata | https://www.wikidata.org/wiki/Q12483 |
| concepts[19].display_name | Statistics |
| concepts[20].id | https://openalex.org/C121332964 |
| concepts[20].level | 0 |
| concepts[20].score | 0.06702357530593872 |
| concepts[20].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[20].display_name | Physics |
| concepts[21].id | https://openalex.org/C151730666 |
| concepts[21].level | 1 |
| concepts[21].score | 0.0 |
| concepts[21].wikidata | https://www.wikidata.org/wiki/Q7205 |
| concepts[21].display_name | Paleontology |
| concepts[22].id | https://openalex.org/C62520636 |
| concepts[22].level | 1 |
| concepts[22].score | 0.0 |
| concepts[22].wikidata | https://www.wikidata.org/wiki/Q944 |
| concepts[22].display_name | Quantum mechanics |
| concepts[23].id | https://openalex.org/C86803240 |
| concepts[23].level | 0 |
| concepts[23].score | 0.0 |
| concepts[23].wikidata | https://www.wikidata.org/wiki/Q420 |
| concepts[23].display_name | Biology |
| keywords[0].id | https://openalex.org/keywords/mathematics |
| keywords[0].score | 0.6396316289901733 |
| keywords[0].display_name | Mathematics |
| keywords[1].id | https://openalex.org/keywords/markov-chain |
| keywords[1].score | 0.6379321813583374 |
| keywords[1].display_name | Markov chain |
| keywords[2].id | https://openalex.org/keywords/attractor |
| keywords[2].score | 0.6033686995506287 |
| keywords[2].display_name | Attractor |
| keywords[3].id | https://openalex.org/keywords/synchronization |
| keywords[3].score | 0.5447115302085876 |
| keywords[3].display_name | Synchronization (alternating current) |
| keywords[4].id | https://openalex.org/keywords/ordinary-differential-equation |
| keywords[4].score | 0.5237728953361511 |
| keywords[4].display_name | Ordinary differential equation |
| keywords[5].id | https://openalex.org/keywords/context |
| keywords[5].score | 0.5026881694793701 |
| keywords[5].display_name | Context (archaeology) |
| keywords[6].id | https://openalex.org/keywords/markov-process |
| keywords[6].score | 0.49864912033081055 |
| keywords[6].display_name | Markov process |
| keywords[7].id | https://openalex.org/keywords/time-reversibility |
| keywords[7].score | 0.47013670206069946 |
| keywords[7].display_name | Time reversibility |
| keywords[8].id | https://openalex.org/keywords/statistical-physics |
| keywords[8].score | 0.4443860352039337 |
| keywords[8].display_name | Statistical physics |
| keywords[9].id | https://openalex.org/keywords/discrete-time-and-continuous-time |
| keywords[9].score | 0.441024512052536 |
| keywords[9].display_name | Discrete time and continuous time |
| keywords[10].id | https://openalex.org/keywords/jump |
| keywords[10].score | 0.4321504533290863 |
| keywords[10].display_name | Jump |
| keywords[11].id | https://openalex.org/keywords/jump-process |
| keywords[11].score | 0.42752930521965027 |
| keywords[11].display_name | Jump process |
| keywords[12].id | https://openalex.org/keywords/applied-mathematics |
| keywords[12].score | 0.3547423481941223 |
| keywords[12].display_name | Applied mathematics |
| keywords[13].id | https://openalex.org/keywords/variable-order-markov-model |
| keywords[13].score | 0.3419708013534546 |
| keywords[13].display_name | Variable-order Markov model |
| keywords[14].id | https://openalex.org/keywords/markov-model |
| keywords[14].score | 0.26474133133888245 |
| keywords[14].display_name | Markov model |
| keywords[15].id | https://openalex.org/keywords/mathematical-analysis |
| keywords[15].score | 0.18507987260818481 |
| keywords[15].display_name | Mathematical analysis |
| keywords[16].id | https://openalex.org/keywords/topology |
| keywords[16].score | 0.16976135969161987 |
| keywords[16].display_name | Topology (electrical circuits) |
| keywords[17].id | https://openalex.org/keywords/differential-equation |
| keywords[17].score | 0.12430712580680847 |
| keywords[17].display_name | Differential equation |
| keywords[18].id | https://openalex.org/keywords/combinatorics |
| keywords[18].score | 0.1205810010433197 |
| keywords[18].display_name | Combinatorics |
| keywords[19].id | https://openalex.org/keywords/statistics |
| keywords[19].score | 0.07646229863166809 |
| keywords[19].display_name | Statistics |
| keywords[20].id | https://openalex.org/keywords/physics |
| keywords[20].score | 0.06702357530593872 |
| keywords[20].display_name | Physics |
| language | en |
| locations[0].id | doi:10.1007/s10884-023-10345-4 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S176689137 |
| locations[0].source.issn | 1040-7294, 1572-9222 |
| locations[0].source.type | journal |
| locations[0].source.is_oa | False |
| locations[0].source.issn_l | 1040-7294 |
| locations[0].source.is_core | True |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | Journal of Dynamics and Differential Equations |
| locations[0].source.host_organization | https://openalex.org/P4310319900 |
| locations[0].source.host_organization_name | Springer Science+Business Media |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310319900, https://openalex.org/P4310319965 |
| locations[0].source.host_organization_lineage_names | Springer Science+Business Media, Springer Nature |
| locations[0].license | cc-by |
| locations[0].pdf_url | |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | https://openalex.org/licenses/cc-by |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Journal of Dynamics and Differential Equations |
| locations[0].landing_page_url | https://doi.org/10.1007/s10884-023-10345-4 |
| indexed_in | crossref |
| authorships[0].author.id | https://openalex.org/A5003876135 |
| authorships[0].author.orcid | https://orcid.org/0000-0002-1406-8052 |
| authorships[0].author.display_name | Maximilian Engel |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Maximilian Engel |
| authorships[0].is_corresponding | False |
| authorships[1].author.id | https://openalex.org/A5023825368 |
| authorships[1].author.orcid | https://orcid.org/0000-0001-9038-0382 |
| authorships[1].author.display_name | Guillermo Olicón-Méndez |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Guillermo Olicón-Méndez |
| authorships[1].is_corresponding | False |
| authorships[2].author.id | https://openalex.org/A5093993444 |
| authorships[2].author.orcid | |
| authorships[2].author.display_name | Nathalie Wehlitz |
| authorships[2].author_position | middle |
| authorships[2].raw_author_name | Nathalie Wehlitz |
| authorships[2].is_corresponding | False |
| authorships[3].author.id | https://openalex.org/A5019520155 |
| authorships[3].author.orcid | https://orcid.org/0000-0002-0114-7819 |
| authorships[3].author.display_name | Stefanie Winkelmann |
| authorships[3].author_position | last |
| authorships[3].raw_author_name | Stefanie Winkelmann |
| 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 | https://doi.org/10.1007/s10884-023-10345-4 |
| open_access.oa_status | hybrid |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Synchronization and Random Attractors in Reaction Jump Processes |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T10621 |
| primary_topic.field.id | https://openalex.org/fields/13 |
| primary_topic.field.display_name | Biochemistry, Genetics and Molecular Biology |
| primary_topic.score | 0.9998000264167786 |
| primary_topic.domain.id | https://openalex.org/domains/1 |
| primary_topic.domain.display_name | Life Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/1312 |
| primary_topic.subfield.display_name | Molecular Biology |
| primary_topic.display_name | Gene Regulatory Network Analysis |
| related_works | https://openalex.org/W2979480265, https://openalex.org/W2019735591, https://openalex.org/W2024318042, https://openalex.org/W2551590868, https://openalex.org/W3106931405, https://openalex.org/W4245666872, https://openalex.org/W2995907685, https://openalex.org/W2081606708, https://openalex.org/W2624969280, https://openalex.org/W1974839821 |
| cited_by_count | 2 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 2 |
| locations_count | 1 |
| best_oa_location.id | doi:10.1007/s10884-023-10345-4 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S176689137 |
| best_oa_location.source.issn | 1040-7294, 1572-9222 |
| best_oa_location.source.type | journal |
| best_oa_location.source.is_oa | False |
| best_oa_location.source.issn_l | 1040-7294 |
| best_oa_location.source.is_core | True |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | Journal of Dynamics and Differential Equations |
| best_oa_location.source.host_organization | https://openalex.org/P4310319900 |
| best_oa_location.source.host_organization_name | Springer Science+Business Media |
| best_oa_location.source.host_organization_lineage | https://openalex.org/P4310319900, https://openalex.org/P4310319965 |
| best_oa_location.source.host_organization_lineage_names | Springer Science+Business Media, Springer Nature |
| best_oa_location.license | cc-by |
| best_oa_location.pdf_url | |
| best_oa_location.version | publishedVersion |
| best_oa_location.raw_type | journal-article |
| best_oa_location.license_id | https://openalex.org/licenses/cc-by |
| best_oa_location.is_accepted | True |
| best_oa_location.is_published | True |
| best_oa_location.raw_source_name | Journal of Dynamics and Differential Equations |
| best_oa_location.landing_page_url | https://doi.org/10.1007/s10884-023-10345-4 |
| primary_location.id | doi:10.1007/s10884-023-10345-4 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S176689137 |
| primary_location.source.issn | 1040-7294, 1572-9222 |
| primary_location.source.type | journal |
| primary_location.source.is_oa | False |
| primary_location.source.issn_l | 1040-7294 |
| primary_location.source.is_core | True |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | Journal of Dynamics and Differential Equations |
| primary_location.source.host_organization | https://openalex.org/P4310319900 |
| primary_location.source.host_organization_name | Springer Science+Business Media |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310319900, https://openalex.org/P4310319965 |
| primary_location.source.host_organization_lineage_names | Springer Science+Business Media, Springer Nature |
| primary_location.license | cc-by |
| primary_location.pdf_url | |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | https://openalex.org/licenses/cc-by |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Journal of Dynamics and Differential Equations |
| primary_location.landing_page_url | https://doi.org/10.1007/s10884-023-10345-4 |
| publication_date | 2024-02-23 |
| publication_year | 2024 |
| referenced_works | https://openalex.org/W172803389, https://openalex.org/W897674100, https://openalex.org/W4295023611, https://openalex.org/W2178233184, https://openalex.org/W20554700, https://openalex.org/W2070430318, https://openalex.org/W913946616, https://openalex.org/W4232753567, https://openalex.org/W2962782145, https://openalex.org/W2990419527, https://openalex.org/W2964301372, https://openalex.org/W2042321087, https://openalex.org/W2085061065, https://openalex.org/W2046451925, https://openalex.org/W2167154952, https://openalex.org/W2981925034, https://openalex.org/W2035364388, https://openalex.org/W4242172554, https://openalex.org/W2067052005, https://openalex.org/W125580901, https://openalex.org/W2062259223, https://openalex.org/W1976443948, https://openalex.org/W2071062848, https://openalex.org/W2063104705, https://openalex.org/W2078119846, https://openalex.org/W2019061200, https://openalex.org/W2964120814, https://openalex.org/W3015638916, https://openalex.org/W1638774395, https://openalex.org/W3135223523, https://openalex.org/W2914908364, https://openalex.org/W2741369543, https://openalex.org/W3011162111, https://openalex.org/W2799151092, https://openalex.org/W2512745720, https://openalex.org/W2112140459, https://openalex.org/W2044043716, https://openalex.org/W3000287905, https://openalex.org/W3156702685 |
| referenced_works_count | 39 |
| abstract_inverted_index.a | 4, 27, 35, 62, 79, 87, 110 |
| abstract_inverted_index.In | 133 |
| abstract_inverted_index.We | 60, 99 |
| abstract_inverted_index.at | 38 |
| abstract_inverted_index.by | 8, 17, 113 |
| abstract_inverted_index.in | 34, 65 |
| abstract_inverted_index.is | 32 |
| abstract_inverted_index.of | 43, 95, 109, 119, 128, 145 |
| abstract_inverted_index.on | 22, 92, 141 |
| abstract_inverted_index.or | 84 |
| abstract_inverted_index.we | 136 |
| abstract_inverted_index.and | 49, 124, 143 |
| abstract_inverted_index.for | 11, 40, 51, 105, 148 |
| abstract_inverted_index.not | 85 |
| abstract_inverted_index.one | 77 |
| abstract_inverted_index.the | 41, 44, 52, 66, 93, 96, 106, 115, 120, 126, 129 |
| abstract_inverted_index.This | 1 |
| abstract_inverted_index.also | 137 |
| abstract_inverted_index.form | 144 |
| abstract_inverted_index.jump | 14, 58 |
| abstract_inverted_index.some | 69 |
| abstract_inverted_index.such | 86 |
| abstract_inverted_index.then | 50 |
| abstract_inverted_index.this | 101, 134 |
| abstract_inverted_index.time | 81 |
| abstract_inverted_index.with | 78 |
| abstract_inverted_index.work | 2 |
| abstract_inverted_index.Based | 21 |
| abstract_inverted_index.chain | 48, 55, 123 |
| abstract_inverted_index.first | 39 |
| abstract_inverted_index.given | 16 |
| abstract_inverted_index.noise | 10 |
| abstract_inverted_index.prove | 100 |
| abstract_inverted_index.sense | 67 |
| abstract_inverted_index.Markov | 13, 47, 54, 122 |
| abstract_inverted_index.common | 9 |
| abstract_inverted_index.delay. | 82 |
| abstract_inverted_index.motion | 118 |
| abstract_inverted_index.occurs | 90 |
| abstract_inverted_index.random | 29, 57, 131, 146, 151 |
| abstract_inverted_index.states | 42 |
| abstract_inverted_index.system | 31 |
| abstract_inverted_index.times. | 59 |
| abstract_inverted_index.Whether | 83 |
| abstract_inverted_index.another | 76 |
| abstract_inverted_index.certain | 80 |
| abstract_inverted_index.depends | 91 |
| abstract_inverted_index.exactly | 74 |
| abstract_inverted_index.general | 139 |
| abstract_inverted_index.induced | 7 |
| abstract_inverted_index.initial | 70, 97 |
| abstract_inverted_index.partial | 102 |
| abstract_inverted_index.process | 112 |
| abstract_inverted_index.provide | 138 |
| abstract_inverted_index.results | 140 |
| abstract_inverted_index.setting | 108 |
| abstract_inverted_index.special | 107 |
| abstract_inverted_index.states. | 98 |
| abstract_inverted_index.uncover | 61 |
| abstract_inverted_index.waiting | 71 |
| abstract_inverted_index.Abstract | 0 |
| abstract_inverted_index.behavior | 89 |
| abstract_inverted_index.chemical | 18 |
| abstract_inverted_index.context, | 135 |
| abstract_inverted_index.embedded | 45, 121 |
| abstract_inverted_index.explores | 3 |
| abstract_inverted_index.reaction | 19 |
| abstract_inverted_index.systems. | 153 |
| abstract_inverted_index.two-step | 36 |
| abstract_inverted_index.analyzing | 114 |
| abstract_inverted_index.augmented | 53 |
| abstract_inverted_index.determine | 125 |
| abstract_inverted_index.dynamical | 30, 152 |
| abstract_inverted_index.existence | 142 |
| abstract_inverted_index.including | 56 |
| abstract_inverted_index.networks. | 20 |
| abstract_inverted_index.processes | 15 |
| abstract_inverted_index.structure | 127 |
| abstract_inverted_index.two-point | 117 |
| abstract_inverted_index.algorithm, | 26 |
| abstract_inverted_index.associated | 130 |
| abstract_inverted_index.attractor. | 132 |
| abstract_inverted_index.attractors | 147 |
| abstract_inverted_index.formulated | 33 |
| abstract_inverted_index.phenomenon | 6 |
| abstract_inverted_index.procedure, | 37 |
| abstract_inverted_index.replicates | 75 |
| abstract_inverted_index.simulation | 25 |
| abstract_inverted_index.stochastic | 24 |
| abstract_inverted_index.time—one | 72 |
| abstract_inverted_index.trajectory | 73 |
| abstract_inverted_index.birth-death | 111 |
| abstract_inverted_index.combination | 94 |
| abstract_inverted_index.that—after | 68 |
| abstract_inverted_index.time-shifted | 63, 103 |
| abstract_inverted_index.Gillespie’s | 23 |
| abstract_inverted_index.corresponding | 28, 116 |
| abstract_inverted_index.discrete-time | 46 |
| abstract_inverted_index.discrete-space | 150 |
| abstract_inverted_index.discrete-time, | 149 |
| abstract_inverted_index.continuous-time | 12 |
| abstract_inverted_index.synchronization | 64, 88, 104 |
| abstract_inverted_index.synchronization-like | 5 |
| cited_by_percentile_year.max | 97 |
| cited_by_percentile_year.min | 95 |
| countries_distinct_count | 0 |
| institutions_distinct_count | 4 |
| citation_normalized_percentile.value | 0.66260816 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |