Perturbations Of Several Planar Integrable Systems With Degenerate Singularity

This dissertation mainly study the perturbations of four planar integrable sys-tems of di?erential equation with degenerate singular point, which are the Li′enardsystem of type (m,n), concretely formed as x = y, y = P(x) +εyQ(x)(deg(P) =m,deg(Q) = n). Whenε= 0, the unperturbed system is Hamiltonian. As m = 3,the first integral is an elliptic Hamiltonian of degree four, it has been studied thor-oughly concerning with the system under small polynomial perturbations, suchas ref.[75–78] for instance. As m = 4, the unperturbed system is a hyperellipticHamiltonian system of degree five. The normal form of hyperelliptic Hamilto-nian of degree five is firstly proposed by I.D.Iliev and L.Gavrilov[58], originallyin order to answer a problem asked by V.I.Arnold[18]. On the other hand, theresearch on the limit cycle for a kind of five-parameter's unfolding of nilpotentcusp with codimension five could be reduced to the problem concerning with limitcycle of a Hyperelliptic Hamiltonian system of degree five under small polynomialperturbations[53]. In this thesis, we consider three kinds of the Hamiltonian vectorfields with an hyperelliptic Hamiltonian of degree five under a class of small poly-nomial perturbations and the quintic perturbation of a quadratic reversible andnon-Hamiltonian system, discuss the estimate to the number of isolated zeros ofthe associated Abelian integrals and the problem about the number of limit cy-cles appearing by some kinds of bifurcations. we get the least upper bound of thenumber of zeros of the associated (pseudo)Abelian integral and the estimate to thelower bound to the global cyclicity of the systems. The thesis is closely related tothe multiple-parameter's unfolding of degenerate singular points with higher order and weakened Hilbert's sixteenth problem.Concretely, we have done the following works.Chapter one is an introduction, we introduce the research backgrounds, pro-gresses about this topic and main works of this paper, some preliminary theoriesand approaches.In Chapter two, we study the quartic polynomial perturbations of a kind ofa hyperelliptic Hamiltonian system of degree five with a homoclinic loop passingthrough a nilpotent saddle, the perturbed system is Li′enard system of type (4,3).We get the maximal order of weak focus is three by applying the Hopf bifurca-tion analysis, and prove that the system under perturbation could have distribu-tion (3,0) of limit cycles. We discuss the perturbation of homoclinic loop passingthrough a nilpotent saddle, obtain the parameter domain for that the system couldbifurcate three limit cycles near the homoclinic loop. On the other hand, we ob-tain the cyclicity of period annulus by applying an algebraic criterion for Abelianintegral which is introduced by M.Grau,et al(Trans.Amer.Math.Soc,363(2011)),through which the problem of extimate of the number of zeros of Abelian integralsis reduced to a purely algebraic problem. By this techniques and approaches, weprove that the isolated number of the zeros of the associated Abelian integral is nomore than four, i.e the vector space generated from Abelian integral is Chebyshevwith accuracy one.In Chapter three, we study the cubic polynomial perturbations of a kind of ahyperelliptic Hamiltonian system of degree five with a degenerate polycycle passingthrough a cusp and a hyperbolic saddle, the perturbed system is the Li′enard systemof type (4,2). We get the maximal order of weak focus is two through making theasymptotic expansion of the first order Melnikov function near the elementarycenter, and proof the family of Abelian integrals is Chebyshev which implies thecyclicity of periodic annulus is two. Furthermore we get the asymptotic expansionof first order Melnikov function near the degenerate polycycle passing a hyperbolic saddle and a cusp of order k in general case, through which we get the lower boundof cyclicity of the degenerate polycycle concerning with the Li′enard system of type(4,2) is two.In Chapter four, we study the cubic polynomial perturbation of a hyperellipticHamiltonian system of degree five with a nilpotent center, which belongs to theLi′enard system of type (4,2). We assert that the system could appear at least twolimit cycles by deducing the asymptotic expansions of first order Melnikov functionnear the nilpotent center and homoclinic loop passing through a hyperbolic saddle.On the other hand, we prove the first order Melnikov function could have at mosttwo zeros on any compact period annulus with the help of the Chebyshev criterionfor Abelian integral proposed by M.Grau, et al and solving the semi-algebraicsystem.In Chapter five, we study the quintic perturbation of a quadratic reversibleand non-Hamiltonian system with an unbounded homoclinic loop, the perturbedsystem is the Li′enard system of type (1,3). The system has an integrating factorformed as the exponential function. With the help of some approaches in realanalysis and following the idea of the integral curves determined by di?erentialequation, we prove that the (pseudo) Abelian integral has the Chebyshev propertyin the geometric view, therefore the cyclicity of the system is one on the finiteplane, which agrees with the Lins-de Melo-Pugh's Conjecture .