{"title":"四元数Julia和Mandelbrot集的新研究使用粘性迭代方法","authors":"Nabaraj Adhikari, Wutiphol Sintunavarat","doi":"10.1016/j.rico.2025.100525","DOIUrl":null,"url":null,"abstract":"<div><div>This paper presents a novel technique for visualizing quaternion Julia and Mandelbrot sets of a quaternion-valued polynomial mapping <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mi>m</mi><mi>q</mi><mo>+</mo><mi>c</mi></mrow></math></span>, where <span><math><mi>q</mi></math></span> is a quaternion variable, <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi><mo>∖</mo><mrow><mo>{</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, and <span><math><mrow><mi>m</mi><mo>,</mo><mi>c</mi></mrow></math></span> are quaternion parameters, by employing the viscosity approximation method. The investigation begins with a study of a new escape criterion, specifically designed for generating quaternion Julia and Mandelbrot sets using the viscosity approximation technique. Based on this result, two dimensions and three dimensions cross-sections of quaternion Julia and Mandelbrot sets are created. The paper also examines how variations in the parameters of the iterative methods impact the resulting sets’ characteristics, such as shape, size, symmetry, and color.</div></div>","PeriodicalId":34733,"journal":{"name":"Results in Control and Optimization","volume":"18 ","pages":"Article 100525"},"PeriodicalIF":3.2000,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A novel investigation of quaternion Julia and Mandelbrot sets using the viscosity iterative approach\",\"authors\":\"Nabaraj Adhikari, Wutiphol Sintunavarat\",\"doi\":\"10.1016/j.rico.2025.100525\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This paper presents a novel technique for visualizing quaternion Julia and Mandelbrot sets of a quaternion-valued polynomial mapping <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mi>m</mi><mi>q</mi><mo>+</mo><mi>c</mi></mrow></math></span>, where <span><math><mi>q</mi></math></span> is a quaternion variable, <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi><mo>∖</mo><mrow><mo>{</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, and <span><math><mrow><mi>m</mi><mo>,</mo><mi>c</mi></mrow></math></span> are quaternion parameters, by employing the viscosity approximation method. The investigation begins with a study of a new escape criterion, specifically designed for generating quaternion Julia and Mandelbrot sets using the viscosity approximation technique. Based on this result, two dimensions and three dimensions cross-sections of quaternion Julia and Mandelbrot sets are created. The paper also examines how variations in the parameters of the iterative methods impact the resulting sets’ characteristics, such as shape, size, symmetry, and color.</div></div>\",\"PeriodicalId\":34733,\"journal\":{\"name\":\"Results in Control and Optimization\",\"volume\":\"18 \",\"pages\":\"Article 100525\"},\"PeriodicalIF\":3.2000,\"publicationDate\":\"2025-01-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Results in Control and Optimization\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2666720725000116\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Results in Control and Optimization","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2666720725000116","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
A novel investigation of quaternion Julia and Mandelbrot sets using the viscosity iterative approach
This paper presents a novel technique for visualizing quaternion Julia and Mandelbrot sets of a quaternion-valued polynomial mapping , where is a quaternion variable, , and are quaternion parameters, by employing the viscosity approximation method. The investigation begins with a study of a new escape criterion, specifically designed for generating quaternion Julia and Mandelbrot sets using the viscosity approximation technique. Based on this result, two dimensions and three dimensions cross-sections of quaternion Julia and Mandelbrot sets are created. The paper also examines how variations in the parameters of the iterative methods impact the resulting sets’ characteristics, such as shape, size, symmetry, and color.