{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:56:52Z","timestamp":1760057812394,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,27]],"date-time":"2025-02-27T00:00:00Z","timestamp":1740614400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we investigate the existence of Hopf bifurcations in a mathematical model that includes economic growth, corruption, and unemployment. The model links these social and economic factors to provide insights into the dynamics of the economy. The motivation for investigating the appearance of Hopf bifurcations is that economic cycles often occur in economics and play an important role. The mathematical model was presented before, but the important topic of the appearance of limit cycles was not investigated. The authors studied some of the effects of corruption on economic growth and unemployment. However, in our work, we focus on the existence of limit cycles. We used different bifurcation parameters to find the conditions for the appearance of Hopf bifurcations. We perform a variety of numerical simulations in which the system presents several Hopf bifurcations. The numerical simulations provide additional support for the theoretical results related to stability and Hopf bifurcations. Finally, we present a discussion and future directions for research.<\/jats:p>","DOI":"10.3390\/axioms14030173","type":"journal-article","created":{"date-parts":[[2025,2,27]],"date-time":"2025-02-27T08:04:44Z","timestamp":1740643484000},"page":"173","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Hopf Bifurcations in a Mathematical Model for Economic Growth, Corruption, and Unemployment: Computation of Economic Limit Cycles"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-7961-8626","authenticated-orcid":false,"given":"Ogochukwu","family":"Ifeacho","sequence":"first","affiliation":[{"name":"Department of Mathematics, New Mexico Tech, Socorro, NM 87801, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5847-678X","authenticated-orcid":false,"given":"Gilberto","family":"Gonz\u00e1lez-Parra","sequence":"additional","affiliation":[{"name":"Department of Mathematics, New Mexico Tech, Socorro, NM 87801, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,27]]},"reference":[{"key":"ref_1","unstructured":"Callen, T. (2012). Gross Domestic Product: An Economy\u2019s All, International Monetary Fund."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1080\/01615440009598956","article-title":"Estimating early American gross domestic product","volume":"33","author":"McCusker","year":"2000","journal-title":"Hist. Methods J. Quant. Interdiscip. Hist."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1038\/s41567-018-0204-y","article-title":"A dynamical systems approach to gross domestic product forecasting","volume":"14","author":"Tacchella","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_4","first-page":"353","article-title":"Modeling the gross domestic product of Nigeria from 1985 to 2018","volume":"6","author":"Makinde","year":"2020","journal-title":"Commun. Stat. Case Stud. Data Anal. Appl."},{"key":"ref_5","first-page":"012138","article-title":"Simulation-dynamic model of long-term economic growth using Solow model","volume":"1353","author":"Boyko","year":"2019","journal-title":"J. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Gonz\u00e1lez-Parra, G., Chen-Charpentier, B., Arenas, A.J., and D\u00edaz-Rodr\u00edguez, M. (2022). Mathematical modeling of physical capital diffusion using a spatial Solow model: Application to smuggling in Venezuela. Economies, 10.","DOI":"10.3390\/economies10070164"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1665","DOI":"10.1016\/j.physa.2009.12.008","article-title":"Stokes integral of economic growth: Calculus and the Solow model","volume":"389","author":"Mimkes","year":"2010","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ani\u0163a, S., Arn\u0103utu, V., and Capasso, V. (2011). An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB, Springer Science & Business Media.","DOI":"10.1007\/978-0-8176-8098-5"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"De la Fuente, A. (2000). Mathematical Methods and Models for Economists, Cambridge University Press.","DOI":"10.1017\/CBO9780511810756"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hritonenko, N., and Yatsenko, Y. (1999). Mathematical Modeling in Economics, Ecology and the Environment, Springer.","DOI":"10.1007\/978-1-4419-9733-3"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/B978-0-444-51676-3.50003-8","article-title":"Models and modelling in economics","volume":"13","author":"Morgan","year":"2012","journal-title":"Philos. Econ."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Radzicki, M.J. (2020). System dynamics and its contribution to economics and economic modeling. System Dynamics: Theory and Applications, Springer.","DOI":"10.1007\/978-1-4939-8790-0_539"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1134\/S2070048218020126","article-title":"Simulation of the \u201cPower\u2013Society\u2013Economics\u201d System with Elements of Corruption Based on Cellular Automata","volume":"10","author":"Stepantsov","year":"2018","journal-title":"Math. Model. Comput. Simul."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"667","DOI":"10.14505\/arle.v11.2(48).37","article-title":"Nonlinear Models Used to Analyze the Relation between Inflation and Unemployment","volume":"11","author":"Ungureanu","year":"2020","journal-title":"J. Adv. Res. Law Econ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"7421","DOI":"10.3934\/math.2021434","article-title":"Stability analysis of an unemployment model with time delay","volume":"6","author":"Petaratip","year":"2021","journal-title":"AIMS Math"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1030","DOI":"10.1016\/j.eap.2022.12.032","article-title":"Does innovation affect the impact of corruption on economic growth? International evidence","volume":"77","author":"Dokas","year":"2023","journal-title":"Econ. Anal. Policy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s101010200045","article-title":"Corruption, economic growth, and income inequality in Africa","volume":"3","year":"2002","journal-title":"Econ. Gov."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/s43069-024-00316-w","article-title":"Modelling and Analysis of the Impact of Corruption on Economic Growth and Unemployment","volume":"Volume 5","author":"Mamo","year":"2024","journal-title":"Operations Research Forum"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"104590","DOI":"10.1016\/j.jedc.2022.104590","article-title":"Cultural persistence in corruption, economic growth, and the environment","volume":"147","author":"Varvarigos","year":"2023","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_20","first-page":"492","article-title":"Investigating the causality between corruption and economic growth: A systematic review","volume":"15","author":"Yusof","year":"2023","journal-title":"Inf. Manag. Bus. Rev."},{"key":"ref_21","first-page":"458","article-title":"The Dynamic Relationship Between Corruption\u2014Inflation: Evidence from Panel Vector Autoregression","volume":"68","author":"Sassi","year":"2017","journal-title":"Jpn. Econ. Rev."},{"key":"ref_22","first-page":"e02069","article-title":"Insight into the optimal control strategies on corruption dynamics using fractional order derivatives","volume":"23","author":"Teklu","year":"2024","journal-title":"Sci. Afr."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"100031","DOI":"10.1016\/j.jeconc.2023.100031","article-title":"The interplay of corruption, human capital, and unemployment in Indonesia: Implications for economic development","volume":"2","author":"Triatmanto","year":"2023","journal-title":"J. Econ. Criminol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2759","DOI":"10.1007\/s11135-022-01481-y","article-title":"Impact of corruption, unemployment and inflation on economic growth evidence from developing countries","volume":"57","author":"Uddin","year":"2023","journal-title":"Qual. Quant."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.mathsocsci.2018.11.001","article-title":"Local bifurcations of three and four-dimensional systems: A tractable characterization with economic applications","volume":"97","author":"Bosi","year":"2019","journal-title":"Math. Soc. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"139","DOI":"10.37193\/CJM.2024.01.10","article-title":"Stability and bifurcation analysis of a four-dimensional economic model","volume":"40","author":"Moza","year":"2024","journal-title":"Carpathian J. Math."},{"key":"ref_27","first-page":"1","article-title":"Bifurcation routes to chaos in an extended Van der Pol\u2019s equation applied to economic models","volume":"2009","author":"Pribylova","year":"2009","journal-title":"Electron. J. Differ. Equ. (EJDE) [Electron. Only]"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Hirsch, M.W., Smale, S., and Devaney, R.L. (2013). Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press.","DOI":"10.1016\/B978-0-12-382010-5.00015-4"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Strogatz, S.H. (2018). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, CRC Press.","DOI":"10.1201\/9780429399640"},{"key":"ref_30","unstructured":"Guckenheimer, J., and Holmes, P. (2013). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Science & Business Media."},{"key":"ref_31","unstructured":"Marsden, J.E., and McCracken, M. (2012). The Hopf Bifurcation and Its Applications, Springer Science & Business Media."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/BF00045469","article-title":"Cyclic motions near a Hopf bifurcation of a four-dimensional system","volume":"3","author":"Balachandran","year":"1992","journal-title":"Nonlinear Dyn."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Fern\u00e1ndez-D\u00edaz, A. (2023). Overview and perspectives of chaos theory and its applications in economics. Mathematics, 12.","DOI":"10.3390\/math12010092"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/173\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:43:33Z","timestamp":1760028213000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/173"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,27]]},"references-count":33,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030173"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030173","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,27]]}}}