Hopf Bifurcation In Liénard Systems By Perturbing A Nilpotent Center

In the theory and application of dynamical systems, Lienard systemis an important model of nonlinear oscillators, which has been widelyand deeply studied by a large of experts. In this thesis, I make a study onthe Hopf bifurcation and homoclinic bifurcation of a concrete Lienardsystem at first, where （0,0） is a nilpotent center of the unperturbedsystem. In discussing Hopf bifurcation near a nilpotent center, I mainlywork on the expansion of the first order Melnikov function near anilpotent center and the calculation method of the coefficients in theexpansion of the first order Melnikov function in the existing literature. Inorder to get the expansion of the first order Melnikov function near anilpotent center more conveniently, I give another method to computethe coefficients b_{l} appearing in the expansion of the first orderMelnikov function of the Lienard system by perturbing a nilpotent center,that is, replacing the coefficients b_{l} by coefficients B_{2l+1}, andgive a lower bound of the number of small-amplitude limit cycles byusing a set of equivalent quantities B_{2l+1} which are able to calculatedirectly. As an application, I investigate some polynomial Lienardsystems, and I obtain concrete results.First, I study the following concrete analytic Lienard system:\begin{equation*}\begin{cases}\dot{x}=y,\\\dot{y}=-x^{3}（1-x）+\varepsilon（a+bx+cx^{2}+x^{3}）y.\end{cases}\end{equation*}I make a study on the Hopf bifurcation and homoclinic bifurcationby the expansion of the first order Melnikov function. Then I obtain a main theorem, that is:（1）\There exists$\varepsilon₀\in （0,\frac{1}{20}）such that\widetilde{M} has at most3zeros on the union（0,\varepsilon₀）\bigcup（\frac{1}{20}-\varepsilon₀,\frac{1}{20}）for all （a,b,c）\in\mathbb{R}^3, and that\widetilde{M} hasprecisely3zeros on each one of the intervals （0,\varepsilon₀）and （\frac{1}{20}-\varepsilon₀,\frac{1}{20}） for some （a,b,c）\in\mathbb{R}^3.（2）\For any0<\mu<\varepsilon₀, there existssome （a,b,c）\in\mathbb{R}^3such that\widetilde{M} has4zeros on the interval（0,\frac{1}{20}）, of which3zeros are in the interval （0,\mu）.Next, I discuss the following Lienard systems:\begin{equation*}\begin{cases}\dot{x}=y-\varepsilon F（x,a）,\\\dot{y}=-g（x,c）.\end{cases}\end{equation*}where a\in R^{n_{1}}, c\in R^{n_{2}}, F and g are C^{\omega}functions, satisfying:F（0,a）=0, g（x,c）=x^{2m-1}（g_{0}+O（x））,m\geq1, g_{0}>0.I study the computation of the coefficients in the expansion of thefirst order Melnikov function of the Li\{e}nard system by perturbing anilpotent center. Through calculating the coefficients B_{1},B_{3},\cdots, B_{2l+1} to replace b_{0}, b_{1},\cdots,b_{l} in theexpansion of the first order Melnikov function, I can discuss the problemof limit cycle bifurcation easily, give a lower bound of the number ofsmall-amplitude limit cycles by using a set of equivalent quantitiesB_{2l+1} which are able to calculate directly.Finally, as an application of the main theorems, I discuss the following polynomial Lienard systems:\begin{equation*}\begin{cases}\dot{x}=y-\varepsilon F（x,a）,\\\dot{y}=-g（x,c）,\end{cases}\end{equation*}where\begin{equation*}F（x,a）=\displaystyle\sum_{j=1}^{n}a_{j}x^{j},\quadg（x,c）=\sum_{j=3}^{n}c_{j}x^{j},\\c₃=1,\\n\geq3.\end{equation*}By the main theorems, we can get the following theorem:let L（n） denote the maximal number oflimit cycles of this system near the origin, then L（3）\geq1,L（4）\geq3, L（5）\geq5, that is, L（n）\geq2n-5for n=3,4,5.