8391

7772

كارشناسي ارشد

رياضي كاربردي

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1392

هشت،83ص.نمودار

ص.ع.به فارسي و انگليسي

حميدرضا ظهوري زنگنه

رسول عاشقي

15/10/92

مجيد گارز، رضا مزروعي

رياضي

ID7772

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

The number of limit cycles for the classical Lienard equations Hadi Hajilari h hajilari@math iut ac ir 2013 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Hamid Reza Zohouri Zangeneh hamidz@cc iut ac ir Advisor Dr Rasoul Asheghi r asheghi@cc iut ac ir 2010 MSC 37G15 34E17 34C07 34C26 Keywords Slow fast system Singular perturbations Limit cycles Relaxation oscillation Classical Li nard equations Abstract The so called Hilbert Smale problem asks for the maximum number of limit cycles thatclassical Li nard equations can have depending on its degree A scalar second order Li nardequation x f x x x 0 can be studied in a phase plane as a system x y F x y x xwhere F x f s ds The degree of a Li nard equation is given by the degree of F In 01976 A Lins W de Melo and C Pugh in 13 conjectured that the maximum number of limit cycles for a classical Li nard equation of degree n would be equal to n 1 the largest integer 2 n 1less than or equal to 2 inducing the occurrence of at most 2 limit cycles in degree 6 and 3limit cycles in degree 7 In 2007 in a paper by F Dumortier D Panazzolo R Roussarie 11 it was shown that more limit cycles can be expected in Li nard equations They have shownthat a Li nard equation of degree 7 with at least 4 limit cycles can exist This easily implied the existence of classical Li nard equations of degree n n 7 with n 1 1 limit cycles 2This showed the conjecture stated by Lins de Melo and Pugh is not true for n 7 In this thesis based on paper by P De Maesschalck and F Dumortier 9 we prove theexistence of classical Li nard equations of degree 6 having 4 limit cycles It implies the existence of a classical Li nard equations of degree n 6 having at least n 1 2 limit 2cycles The counterexamples prove to occur in systems x y 2 x2 5 x3 35 x4 12 x5 21 x6 1 46 46 y b x de ned for 0 0 and 0 0 for some su ciently small 0 0 and 0 0 All theselimit cycles are hyperbolic and surround a hyperbolic focus that is attracting when 0and repelling when 0 Also these limit cycles are relaxation oscillations in the sense thatthe speed close to the fast orbit is of order O 1 while the speed near the slow curves is oforder O The relaxation oscillation itself is of size O 1 This improves the counterexample

حميدرضا ظهوري زنگنه

رسول عاشقي

مجيد گارز، رضا مزروعي