{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,17]],"date-time":"2026-10-17T04:27:11Z","timestamp":1792211231917,"version":"4.4.0"},"reference-count":26,"publisher":"AIP Publishing","issue":"12","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2022,12,1]]},"abstract":"<jats:p>In this paper, an origami structure of period-1 motions to spiral homoclinic orbits in parameter space is presented for the R\u00f6ssler system. The edge folds of the origami structure are generated by the saddle-node bifurcations. For each edge, there are two layers to form the origami structure. On one layer of the origami structure, there is a pair of period-doubling bifurcations inducing periodic motions from period-1 to period-2n motions (n=1,2,\u2026,\u221e). On such a layer, the unstable period-1 motion goes to the homoclinic orbits with a mapping eigenvalue approaching negative infinity. However, on the corresponding adjacent layers, no period-doubling bifurcations exist, and the unstable period-1 motion goes to the homoclinic orbit with a mapping eigenvalue approaching positive infinity. To determine the origami structure of the period-1 motions to homoclinic orbits, the implicit map of the R\u00f6ssler system is developed through the discretization of the corresponding differential equations. The Poincar\u00e9 mapping section can be selected arbitrarily. Before construction of the origami structure, the bifurcation diagram of periodic motions varying with one parameter is developed, and trajectories of stable periodic motions on the bifurcation diagram to homoclinic orbits are illustrated. Finally, the origami structures of period-1 motions to homoclinic orbits are developed through a few layers. This study provides the mathematical mechanisms of period-1 motions to homoclinic orbits, which help one better understand the complexity of periodic motions near the corresponding homoclinic orbit. There are two types of infinitely many homoclinic orbits in the R\u00f6ssler system, and the corresponding mapping structures of the homoclinic orbits possess positive and negative infinity large eigenvalues. Such infinitely many homoclinic orbits are induced through unstable periodic motions with positive and negative eigenvalues accordingly.<\/jats:p>","DOI":"10.1063\/5.0131970","type":"journal-article","created":{"date-parts":[[2022,12,9]],"date-time":"2022-12-09T16:58:00Z","timestamp":1670605080000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["On an origami structure of period-1 motions to homoclinic orbits in the R\u00f6ssler system"],"prefix":"10.1063","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4502-3006","authenticated-orcid":false,"given":"Siyuan","family":"Xing","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, California Polytechnic State University 1 , San Luis Obispo, California 93407, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8208-6108","authenticated-orcid":false,"given":"Albert C. J.","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Mechatronics Engineering, Southern Illinois University Edwardsville 2 , Edwardsville, Illinois 62026-1805, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2022,12,9]]},"reference":[{"key":"2023080922114618900_c1","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1016\/0375-9601(76)90101-8","article-title":"An equation for continuous chaos","volume":"57","year":"1976","journal-title":"Phys. Lett. A"},{"key":"2023080922114618900_c2","first-page":"1664","article-title":"Different types of chaos in two simple differential equations","volume":"31","year":"1976","journal-title":"Z. Nat. 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