سيكل هايﻫﺎي ﺣﺪي در اﺧﺘﻼﻻت ﮐﻮﭼﮏ ﯾﮏ ﺳﯿﺴﺘﻢ ﻫﻤﯿﻠﺘﻮﻧﯽﻗﻄﻌﻪ اي ﺧﻄﯽ ﻣﺴﻄﺢ ﺑﺎ ﯾﮏ ﺧﻂ ﺟﺪاﺳﺎز نامنظم

چكيده انگليسي :

Limit cycles in small perturbations of a planar piecewise linear Hamiltonian system with a non regular separation line VALIOLLAH MOHAMADI valiollah mohammadi73@gmail com January 26 2020 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Rasoul Asheghi r asheghi@cc iut ac irrefereeA Dr Majid Gazor mgazor@cc iut ac irrefereeB Dr Reza Mazrooei mazrooei@cc iut ac ir2018 MSC 34C05 34C07 37G15Abstract In the past few decades many studies have been devoted to the investigation of limit cycles of planar picewisesmooth dynamical system defined on two zones separated by a line see e g 2 to 20 Commonly the separationline of the two zones is straight In this case using numerical simulation Huan and Yang 8 provided an exampleillustrating the existence of three limit cycles for piecewise linear systems Rigorous proofs of the existence of theselimit cycles were given in 9 10 In 11 Chen and Du constructed a quadratic system which can have nine smalllimit cycles bifurcated from a center Perturbing quadratic systems with quadratic and cubic polynomial functionsXiong 12 and Tian and Yu 13 found six and ten limit cycles respectively For piecewise polynomial nearHamiltonian systems of degree n the number of limit cycles was studied in 14 16 When the separation line isnot a stright one more than three limit cycles can be found for planar piecewise linear systems with two regions For examples by constructing some broken lines as the boundary between the two linear zones Braga and Mollo 17 proved that 4 5 6 or 7 limit cycles can apear Two more concrete systems with piecewise continuousseparation lines were given in 18 where Corollaries 3 and 5 con rm the following conjecture proposed in 17 For a given n N there is a piecewise linear system with two zones in the plane with exactly n limit cycles Recently Cardin and Torregrosa 19 considered a planar piecewise linear system de ned in two angular zonesseparated by x y x 0 y 0 x y y tan x y 0 0 2 x y x 0 y 0 x y x 0 y 0 2Using higher order Melnikov function method the authors showed the existence of ve limit cycles for such systemup to a sixth order perturbation in To distinguish from the case of straight lines if a separation line is piecewisesmooth we call it a non regular one It is pointed out in 17 that non regularity in the separation lines plays animportant role in studying of the number of limit cycles for piecewise systems This motivates us to consider thefollowing problems a For planar polynomial differential systems of degree n what is the maximal number of limit cycles produced byperturbing a planar piecewise linear system with a period annuls around the origin up to the rst order in if thepiecewise period annuls is divided by 0 b Does the maximal number of limit cycles depend on the angle 0 To study these problems in this thesis we rst construct a planar piecewise near Hamiltonian system of degree n whose unperturbed system is piecewise linear and divided by 0 such that for any 0 theunperturbed system has a family of periodic orbits between the origin and a homoclinic loop around the origin In 14 the authors also considered a planar piecewise linear Hamiltonian system with two zones perturbed by nth degree polynomials The main difference between the unperturbed systems constructed in this thsis and the onein 14 is the non regularity in the separation lines where in 14 it is a straight line and in this thesis it is anon regular one 0 Then using the rst order Melnikov function we investigate the maximal

محمدي، ولي اله

سيكل هايﻫﺎي ﺣﺪي در اﺧﺘﻼﻻت ﮐﻮﭼﮏ ﯾﮏ ﺳﯿﺴﺘﻢ ﻫﻤﯿﻠﺘﻮﻧﯽﻗﻄﻌﻪ اي ﺧﻄﯽ ﻣﺴﻄﺢ ﺑﺎ ﯾﮏ ﺧﻂ ﺟﺪاﺳﺎز نامنظم

سيكل حدي , تابع ملنيكوف , سيستم چبيشف , حلقه هموكلينيك