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رسول عاشقی حسین آبادی (استاد راهنما) رسول کاظمی نجف آبادی (استاد مشاور) حمیدرضا ظهوری زنگنه (استاد مشاور)
Ali Bakhshalizadeh Badaki
علی بخشعلی زاده بادکی


دانشکده ریاضی
Doctor of Philosophy (PhD)


Limit cycles and monotonicity of Abelian integrals in some smooth and non-smooth systems
Dynamical systems theory is a powerful tool to analyze and understand the dynamical evolutions of a diverse range of problems in natural sciences and engineering . Most of the problems that have been considered so far were systems defined by a smooth function of its argument and there is a well-developed theory to analyze such systems qualitatively and geometrically . This approach has been proved to be very helpful for understanding many nonlinear phenomena in mathematics , physics , engineering , biology , etc . However , many significant phenomena are not smooth which occur everywhere such as electrical circuits that have switches , problems with friction and models in the social and financial sciences that a continuous change can cause discrete responses \\cite{Barbashin, Bernardo ,Ito,Filippov,Kunze} . Such problems are characterized by piecewise-smooth functions . They have a very rich dynamics and underlying mathematical structure . By making careful mathematical assumption consistent with the underlying physical assumption of the problem , it is possible to extend naturally many of the concepts once assumed to be in the domain of smooth dynamical systems . One of the important topics in the \\begin{equation}\\label{bd1} \\dot{x}=X_{n}(x,y), \\qquad \\dot{y}=Y_{n}(x,y) ,\ag{1} \\end{equation} where $ X_n $ and $ Y_n $ are polynomials of degree $ n $ . To date, this problem remains unsolved even for the quadratic case, i.e. $ n = 2 $ . We note that a linear system may have periodic orbits but has no limit cycle. In \\cite{VArnold} , Arnold proposed a weaker version of the problem by restricting \\eqref{bd1} to the form \\begin{equation}\\label{bd2} \\frac{dx}{dt}=-{\\partial H(x,y)\\over \\partial y}+\\varepsilon f(x,y),\\qquad \\frac{dy}{dt}={\\partial H(x,y)\\over \\partial x}+\\varepsilon g(x,y) ,\ag{2} \\end{equation} where $ \\varepsilon $ is a small parameter, $ H $ is a polynomial of degree $ n + 1 $ and $ f $ and $ g $ are polynomials of degree less than or equal to $ n $ . Thus, \\eqref{bd2} is a perturbation of the Hamiltonian system \\begin{equation}\\label{bd3} \\frac{dx}{dt}=-{\\partial H(x,y)\\over \\partial y},\\qquad \\frac{dy}{dt}={\\partial H(x,y)\\over \\partial x} .\ag{3} \\end{equation} To tackle the Hilbert 16th problem for \\eqref{bd2}, besides the Hopf bifurcation and homoclinic (or heteroclinic) bifurcation analysis, a crucial step is to study the cyclicity of period annulus of $ \\X_H $ which is roughly the total number of limit cycles (counting multiplicity) that can be bifurcated from a period annulus (or annuli) of \\eqref{bd3}. On the other side, the study of the number of limit cycles for discontinuous differential systems in the plane can be seen as an extension of the Hilbert 16th problem . But the n aive ideas of extending Hilbert’s 16th problem to non-smooth systems should be discarded since it was shown in $ 2015 $ that infinitely limit cycles are possible even in a piecewise linear system \\cite{Llibre1} . Because of the importance of this problem, in this thesis, we investigate the limit cycles and monotonosity of Abelian integrals in some smooth and non-smooth systems that plays an important role in the dynamic analysis of such system s .
یکی از مهمترین مسائل در نظریه کلاسیک دستگاه‌های دینامیکی مسأله‌ی شانزدهم هیلبرت است. برای حل این مسأله علاوه بر تحلیل انشعاب هاپف و انشعاب هموکلینیک (یا هتروکلینیک)، یک گام حیاتی مطالعه‌ی سیکل‌پذیری طوق تناوبی میدان برداری همیلتونی است که تقریبا برابر با تعداد کل سیکل‌های حدی (با احتساب تکرار) است که می‌توانند از طوق تناوبی (یا طوق‌های تناوبی) منشعب شوند. از طرفی مطالعه‌ی تعداد سیکل‌های حدی دستگاه‌های دیفرانسیل ناپیوسته در صفحه می‌تواند به عنوان توسیعی از این مسأله دیده شود. به دلیل اهمیت این مسأله، در این رساله به بررسی و مطالعه سیکل‌های حدی و یکنوایی انتگرال‌های آبلی در دستگاه‌های هموار و قطعه‌ای هموار خواهیم پرداخت که نقش مهمی در تحلیل دینامیک چنین دستگاه‌هایی دارند.

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