{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T21:21:30Z","timestamp":1768252890011,"version":"3.49.0"},"reference-count":30,"publisher":"World Scientific Pub Co Pte Lt","issue":"16","funder":[{"name":"JSPS","award":["18K04380\/18K18874\/19K15108"],"award-info":[{"award-number":["18K04380\/18K18874\/19K15108"]}]},{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia","award":["PTDC\/EGE-ECO\/30080\/2017"],"award-info":[{"award-number":["PTDC\/EGE-ECO\/30080\/2017"]}]},{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia","award":["CEECIND\/02741\/2017"],"award-info":[{"award-number":["CEECIND\/02741\/2017"]}]},{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia","award":["UIDB\/04105\/2020"],"award-info":[{"award-number":["UIDB\/04105\/2020"]}]},{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia","award":["UIDB\/00731\/202"],"award-info":[{"award-number":["UIDB\/00731\/202"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2020,12,30]]},"abstract":"<jats:p> This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group [Formula: see text] [Elmhirst, 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models confirms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics. <\/jats:p>","DOI":"10.1142\/s0218127420502405","type":"journal-article","created":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T09:16:23Z","timestamp":1609319783000},"page":"2050240","source":"Crossref","is-referenced-by-count":10,"title":["Breaking and Sustaining Bifurcations in SN-Invariant Equidistant Economy"],"prefix":"10.1142","volume":"30","author":[{"given":"H.","family":"Aizawa","sequence":"first","affiliation":[{"name":"Department of Civil and Environmental Engineering, Tohoku University, Aoba, Sendai 980-8579, Japan"}]},{"given":"K.","family":"Ikeda","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Tohoku University, Aoba, Sendai 980-8579, Japan"}]},{"given":"M.","family":"Osawa","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Tohoku University, Aoba, Sendai 980-8579, Japan"}]},{"given":"J. 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