سیستم‌های دینامیکی و نظریه آشوب سیستم‌های دینامیکی و نظریه آشوب

توضیحات

سیستم‌های دینامیکی شاخه‌هایی گسترده از دانش ریاضی و کاربردهای آن را در بر گرفته و به عنوان یکی از زمینه‌های فعال و زنده آن مطرح است. بیشتر از سه قرن پیش نیوتن بذر این علم را کاشته است و در حدود یک قرن پیش هنری پوانکاره، این شاخه از علم را به درختی تناور و محکم تبدیل کرد. به واسطه تحلیل یک مدل خاص در یک مسیله طبیعی یا ریاضی، تعاریف و صورت‌بندی‌های مختلفی در این شاخه ایجاد شده است. به عنوان مثال از میان تعاریف بسیار، چهار تعریف مهم از آشوبناک بودن یک سیستم دینامیکی در این کتاب ارایه شده‌اند. با توجه به دلایل یاد شده، دسته‌بندی‌های مختلفی از انواع سیستم‌های دینامیکی مطرح است؛ مانند سیستم‌های دینامیکی گسسته و سیستم‌های دینامیکی پیوسته، سیستم‌های متناهی‌البعد در مقابل نامتناهی‌البعد و غیره. سیستم‌های دینامیکی مورد بحث در کتاب حاضر از نوع سیستم‌های دینامیکی گسسته و مشتق‌پذیر می‌باشد و برخی از مباحث آن نیز در زمره دینامیک توپولوژیک به ‌شمار ‌می‌آیند. این کتاب، در عین‌ حال که مباحث متنوعی را در زمینه‌های یادشده عرضه ‌می‌کند، بر محور آشوب می‌چرخد و این پدیده را با جزییاتی بسیار در دستگاه‌های دینامیکی مختلف مانند نگاشت‌ها روی فضاهای انتقال، بازه‌های حقیقی، چنبره و جاذب‌های عجیب مانند جاذب نعل ‌‌اسب و جاذب سیم‌پیچ و همچنین در اتوماتاهای سلولی بررسی ‌می‌کند. در این کتاب چهار تعریف مختلف از آشوب ارایه‌ شده ‌است که به‌ترتیب عبارت‌اند ‌از: 1- آشوب به معنای دوانی؛ 2- مثبت‌بودن نمای لیاپانوف؛ 3- مثبت‌بودن آنتروپی توپولوژیکی و 4- مثبت‌بودن نرخ رشد نمایی مدارهای تناوبی. چنانکه پیشتر گفته شد یکی از موارد اهمیت سیستم‌های دینامیکی، توانمند بودن این رشته برای مدل‌سازی پدیده‌های طبیعی است. امروزه ریاضی‌دان‌های بسیاری از این رشته برای مدل‌سازی پدیده‌های طبیعی در شاخه‌هایی از دانش مانند هواشناسی، زمین‌شناسی، انتقال جرم و حرارت، مدارهای ماهواره‌ای، مکانیک سماوی و نجوم، دریاشناسی و مکانیک سیالات، گرانش و کیهان‌شناسی استفاده می‌کنند. این کتاب به‌عنوان یک متن خودآموز درسی برای دوره‌های کارشناسی و کارشناسی ارشد رشته ریاضیات تالیف ‌شده ‌است، اما با توجه به کاربردهای این رشته، دیگر رشته‌های مرتبط نیز می‌توانند از این منبع استفاده کنند.,

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حال که مباحث متنوعی را در زمینه&zwnj;های یادشده عرضه &zwnj;می&zwnj;کند، بر محور آشوب می&zwnj;چرخد و این پدیده را با جزییاتی بسیار در دستگاه&zwnj;های دینامیکی مختلف مانند نگاشت&zwnj;ها روی فضاهای انتقال، بازه&zwnj;های حقیقی، چنبره و جاذب&zwnj;های عجیب مانند جاذب نعل &zwnj;&zwnj;اسب و جاذب سیم&zwnj;پیچ و همچنین در اتوماتاهای سلولی بررسی &zwnj;می&zwnj;کند. در<span lang=\"AR-SA\" style=\"font-size: 14.0pt; line-height: 150%; font-family: 'B Zar'; mso-ascii-font-family: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;\">&lrm;</span><span lang=\"AR-SA\" style=\"font-size: 14.0pt; line-height: 150%; font-family: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: 'B Zar'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;\"> </span>این کتاب چهار تعریف مختلف از آشوب ارایه&zwnj; شده &zwnj;است که به&zwnj;ترتیب عبارت&zwnj;اند &zwnj;از: <span dir=\"RTL\" lang=\"FA\" style=\"font-size: 14.0pt; 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