{"title":"曼德勃洛特球茎的大小","authors":"A.C. Fowler , M.J. McGuinness","doi":"10.1016/j.csfx.2019.100019","DOIUrl":null,"url":null,"abstract":"<div><p>We provide an analytic estimate for the size of the bulbs adjoining the main cardioid of the Mandelbrot set. The bulbs are approximate circles, and are associated with the stability regions in the complex parameter <em>μ</em>-space of period-<em>q</em> orbits of the underlying map <span><math><mrow><mi>z</mi><mo>→</mo><msup><mi>z</mi><mn>2</mn></msup><mo>−</mo><mi>μ</mi></mrow></math></span>. For the (<em>p, q</em>) orbit with winding number <em>p</em>/<em>q</em>, the associated stability bulb is an approximate circle with radius <span><math><mrow><mstyle><mfrac><mn>1</mn><msup><mi>q</mi><mn>2</mn></msup></mfrac></mstyle><mi>sin</mi><mstyle><mfrac><mrow><mi>π</mi><mi>p</mi></mrow><mi>q</mi></mfrac></mstyle></mrow></math></span>.</p></div>","PeriodicalId":37147,"journal":{"name":"Chaos, Solitons and Fractals: X","volume":"3 ","pages":"Article 100019"},"PeriodicalIF":0.0000,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.csfx.2019.100019","citationCount":"3","resultStr":"{\"title\":\"The size of Mandelbrot bulbs\",\"authors\":\"A.C. Fowler , M.J. McGuinness\",\"doi\":\"10.1016/j.csfx.2019.100019\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We provide an analytic estimate for the size of the bulbs adjoining the main cardioid of the Mandelbrot set. The bulbs are approximate circles, and are associated with the stability regions in the complex parameter <em>μ</em>-space of period-<em>q</em> orbits of the underlying map <span><math><mrow><mi>z</mi><mo>→</mo><msup><mi>z</mi><mn>2</mn></msup><mo>−</mo><mi>μ</mi></mrow></math></span>. For the (<em>p, q</em>) orbit with winding number <em>p</em>/<em>q</em>, the associated stability bulb is an approximate circle with radius <span><math><mrow><mstyle><mfrac><mn>1</mn><msup><mi>q</mi><mn>2</mn></msup></mfrac></mstyle><mi>sin</mi><mstyle><mfrac><mrow><mi>π</mi><mi>p</mi></mrow><mi>q</mi></mfrac></mstyle></mrow></math></span>.</p></div>\",\"PeriodicalId\":37147,\"journal\":{\"name\":\"Chaos, Solitons and Fractals: X\",\"volume\":\"3 \",\"pages\":\"Article 100019\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.csfx.2019.100019\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Chaos, Solitons and Fractals: X\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S259005441930017X\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Chaos, Solitons and Fractals: X","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S259005441930017X","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
We provide an analytic estimate for the size of the bulbs adjoining the main cardioid of the Mandelbrot set. The bulbs are approximate circles, and are associated with the stability regions in the complex parameter μ-space of period-q orbits of the underlying map . For the (p, q) orbit with winding number p/q, the associated stability bulb is an approximate circle with radius .
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