6326

5914

كارشناسي ارشد

رياضي كاربردي

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1390

[هشت]،106ص.: مصور

ص.ع.به فارسي و انگليسي

حميدرضا ظهوري زنگنه

رسول عاشقي

8/8/90

فريد بهرامي، رضا مزروعي

رياضي

ID5914

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

Limit Cycle Bifurcations by Perturbing a Cuspidal Loop in a Hamiltonian System Ali KochakiZefreh a kochakizefreh@math iut ac ir 13 August 2011 Master of Science Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr HamidReza ZohouriZangeneh hamidz@cc iut ac ir Advisor Dr Rasoul Asheghi rasoul asheghi@cc iut ac ir Department Graduate Program Coordinator Dr Azam Etemad ae110mat@cc iut ac ir 2000 MSC 34C05Key words Cuspidal loop Melnikov function Limit cycle Bifurcation Hamiltonian sys tem Z3 equivariance Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Abstract In 1901 at the International Congress of Mathematicians in Paris David Hilbert posed23 mathematical problems of which the second part of the 16th one is to nd the maximalnumber and relative positions of limit cycles of planar polynomial di erential systems ofdegree n x Pn x y y Qn x y There are many works on nding the maximal number of limit cycles and raising the lowerbound of Hilbert number H n for general planar polynomial systems or for individual sys tems with certain degree but the problem is still open even for the quadratic case n 2 A detailed history of published work and related literatures can be found in Li 2003 Ilyashenko 2002 and Han 2006 Many studies had been done for planar systems closeto Hamiltonian systems especially for quadratic and cubic systems The main results are onthe number of limit cycles which appear near a center a periodic oval or a homoclinic loop byperturbations The rst order Melnikov function which is called also Abelian integral playsan important role in getting these results There have been some interesting studies on thebifurcation of limit cycles of equivariant polynomial systems See Yu and Han 2005 haveshown that Z2 equivariant cubic systems can have 12 small amplitude limit cycles and Yu Han and Yaun 2006 have shown that Z3 equivariant cubic systems can have 4 limit cycles This thesis is divided into two main parts In the rst part which is based on an article byMaoan Han Hong Zang and Junmin Yang Limit cycle bifurcations by perturbing a cuspidalloop in a Hamiltonian system 2009 we study the analytical property of the rst Melnikovfunction for general Hamiltonian systems exhibiting a cuspidal loop of order m and obtain itsasymptotic expansion at the Hamiltonian value corresponding to the loop Then by using the rst coe cients of this expansion we give some conditions for the perturbed system to have4 5 or 6 limit cycles in a neighborhood of the loop As an application of our main results weconsider some polynomial Lienard systems of order three and nd 4 5 or 6 limit cycles Inthe second part which based on an article by Hongyan Ma and Maoan Han Limit cycles of aZ3 equivariant near Hamiltonian system 2009 we have studied the number of limit cycles 1

حميدرضا ظهوري زنگنه

رسول عاشقي

فريد بهرامي، رضا مزروعي