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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 57382 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Razvan, Mohammad Reza
- Abstract:
- Chaos in dynamical systems refers to the presence of extreme sensitivity to initial conditions in the system’s behavior, which renders the long-term prediction of the sys- tem’s behavior either impossible or highly challenging. This chaotic behavior is also present in deterministic systems. The observation of this concept, besides philosophical changes it has brought about, has been scrutinized in various domains and has provided an explanation for many observed physical, chemical, and biological phenomena. This phenomenon has been studied in diverse sectors such as fluid dynamics, neuronal dy- namics, and chemical reactions. In this research, we intend to analyze chaotic dynamical systems using metrics such as topological entropy, Lyapunov exponents, and Hausdorff dimension, and we will introduce criteria for identifying these systems. To comprehend these metrics, we will initially explore the foundations of chaos theory and the concept of entropy, followed by the introduction and analysis of several recognized dynamical models across different fields. In this section, through the examination of methodologies employed in scholarly articles, we will analyze the estimation of chaos in each model and endeavor to review the results of each through numerical algorithms, subsequently comparing the outcomes. Finally, we will discuss processes that induce chaos in systems and examine the applications of these processes
- Keywords:
- Dynamical Systems ; Chaos Theory ; Entropy ; Lyapunov Exponent ; Hausdorff Di- Mension ; Topological Entropy ; Chaotic Behavior
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- مقدمه‌ای بر نظریه‌ی آشوب
- پیشینه
- انواع آشوب
- آشوب از دیدگاه دِونی
- آشوب از دیدگاه لیاپانوف
- آشوب و آنتروپی
- آنچه از آشوب به یاد داشته باشیم!
- جاذب‌های پیچیده، فراکتال‌ها و بعد
- جاذب‌ها
- فراکتال‌ها
- کمیت‌های آشوب
- نمای لیاپانوف
- پایداری جواب‌های تناوبی: نظریه‌ی فلوکه
- نمای لیاپانوف برای نگاشت‌های یک بعدی
- نمای لیاپانوف برای نگاشت‌های n-بعدی
- حل عددی نماهای لیاپانوف
- آنتروپی
- آنتروپی توپولوژیک
- آنتروپی برمبنای نظریه‌ی اندازه
- قانون تغییراتی آنتروپی
- آنتروپی و نمای لیاپانوف
- نمای لیاپانوف
- تشخیص آشوب
- آشوب و خمینه‌های پایدار و ناپایدار
- آشوب در اثر انشعاب
- انشعاب و آشوب
- آشوب در اثر نوسانات ناگهانی
- نوع I
- نوع II
- نوع III
- نوع V
- نوع X
- آرام-ناآرام
- چالش‌های عددی
- مراجع