{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:40:51Z","timestamp":1760143251781,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T00:00:00Z","timestamp":1706572800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The dynamical systems of trigonometric functions are explored, with a focus on sz=sin(z) and the fractal image created by iterating the Newton map, Fs(z), of s(z). The basins of attraction created from iterating Fs(z) are analyzed, and some bounds are determined for the primary basins of attraction. We further prove x and y-axis symmetry of the Newton map as well as some interesting results on periodic points on the real axis.<\/jats:p>","DOI":"10.3390\/sym16020162","type":"journal-article","created":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T05:14:32Z","timestamp":1706591672000},"page":"162","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dynamics of Iterations of the Newton Map of sin(z)"],"prefix":"10.3390","volume":"16","author":[{"given":"Aim\u00e9e","family":"Cloutier","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Rose-Hulman Institute of Technology, Terre Haute, IN 47803, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jerry","family":"Dwyer","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Mississippi, Oxford, MS 38677, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roger W.","family":"Barnard","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"William D.","family":"Stone","sequence":"additional","affiliation":[{"name":"Department of Mathematics, New Mexico Institute of Technology, Socorro, NM 87801, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9199-7111","authenticated-orcid":false,"given":"G. Brock","family":"Williams","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0097-8493(94)90097-3","article-title":"Newton\u2019s Method for multiple roots","volume":"18","author":"Gilbert","year":"1994","journal-title":"Comput. Graph."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1080\/07468342.1995.11973661","article-title":"The Dynamics of Newton\u2019s Method for Cubic Polynomials","volume":"26","author":"Walsh","year":"1995","journal-title":"Coll. Math. J."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Alexander, D., Iavernaro, F., and Rosa, A. (2012). Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable during 1906\u20131942, American Mathematical Society.","DOI":"10.1090\/hmath\/038"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/BF00047501","article-title":"Newton\u2019s method and complex dynamical systems","volume":"13","author":"Haeseler","year":"1988","journal-title":"Acta Appl. Math."},{"key":"ref_5","first-page":"139","article-title":"The Dynamics of Newton\u2019s Method","volume":"Volume 49","author":"Devaney","year":"1994","journal-title":"Proceedings of the Symposia in Applied Mathematics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1666","DOI":"10.1080\/17476933.2018.1547285","article-title":"Conjugacies of the Newton Maps of Rational Functions","volume":"64","author":"Barnard","year":"2019","journal-title":"Complex Var. Elliptic Equ."},{"key":"ref_7","first-page":"9","article-title":"Iteration of Complex Functions and Newton\u2019s Method","volume":"23","author":"Dwyer","year":"2009","journal-title":"Aust. Sr. Math. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1853467","DOI":"10.1155\/2020\/1853467","article-title":"Fixed Points, Symmetries, and Bounds for Basins of Attraction of Complex Trigonometric Functions","volume":"2020","author":"Bray","year":"2020","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_9","unstructured":"Saff, E.B., and Snider, A.D. (2003). Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Prentice Hall. [3rd ed.]."},{"key":"ref_10","unstructured":"Stankewitz, R. (2009). Explorations in Complex Analysis, Mathematical Association of America."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Conway, J. (1978). Functions of One Complex Variable I, Springer.","DOI":"10.1007\/978-1-4612-6313-5"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1080\/00029890.1975.11994008","article-title":"Period three implies chaos","volume":"82","author":"Li","year":"1975","journal-title":"Am. Math. Mon."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/162\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:51:42Z","timestamp":1760104302000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/162"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,30]]},"references-count":12,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["sym16020162"],"URL":"https:\/\/doi.org\/10.3390\/sym16020162","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,1,30]]}}}