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In solving real world systems for higher-codimension bifurcation problems, one often faces the difficulty in computing the normal form or the focus values associated with generalized Hopf bifurcation, and the normal form with unfolding for higher-codimension Bogdanov-Takens bifurcation. The difficulty is not only coming from the teadious symbolic computation of focus values, but also due to the restriction on the system parameters, which frequently leads to failure of the conventional approach used in the computation even for simple $2$-dimensional nonlinear dynamical systems. In this paper, we use a simple 2-dimensional epidemic model, for which the conventional approach fails in analyzing the stability of limit cycles arising from Hopf bifurcation, to illustrate how our method can be efficiently applied to determine the codimension of Hopf bifurcation. Further, we apply the simplest normal form theory to consider codimension-3 Bogdanov-Takens bifurcation and present an efficient one-step transformation approach, compared with the classical six-step transformation approach to demonstrate the advantage of our method., Comment: 36 pages, 11 figures

In the study of the number of limit cycles of near-Hamiltonian systems, the first order Melnikov function plays an important role. This paper aims to establish a development of a known method to estimate the upper bound of the number of zeros of the function., Comment: This paper has been accepted for publication in Science China Mathematics

In this paper, by means of the Melnikov functions we consider bifurcations of harmonic or subharmonic solutions from a periodic solution of a planar Hamiltonian system under impulsive perturbation. We give some sufficient conditions under which a harmonic or subharmonic solution exists., Comment: This paper was submitted to Nonlinear Analysis: Theory, Methods & Applications

In this paper, we study the analytical property of the Poincare return map and the generalized focal values of an analytical planar system with a nilpotent focus or center. Then we use the focal values and the map to study the number of limit cycles of this kind of systems with parameters, and obtain some new results on the lower and upper bounds of the maximal number of limit cycles near the nilpotent focus or center., Comment: This paper was submitted to Journal of Mathematical Analysis and Applications

Lienard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Lienard systems of arbitrary degree on the plane, and develop a new method to obtain a lower bound of the maximal number of limit cycles. Using the method and basing on some known results for lower degree we obtain new estimations of the number of limit cycles in the systems which greatly improve existing results., Comment: This paper was submitted to Nonlinear Analysis: Real World Applications

In this paper, we give a positive answer to the open question: Can there exist 4 limit cycles in quadratic near-integrable polynomial systems? It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least 4 limit cycles with (3,1) distribution. The method of Melnikov function is used., Comment: 42 pages, 10 figures