Abstract:
This paper contains two parts. In the first part, we shall show that the result given in the Zoladek's example [1], which claims the existence of 11 small-amplitude limit cycles around a singular point in a particular cubic vector filed, is not correct. Mistakes made in the paper [1] leading to the erroneous conclusion have been identified. In fact, only 9 small-amplitude limit cycles can be obtained from this example after the mistakes are corrected, which agrees with the result obtained later by using the method of focus value computation [2]. In the second part, we present an example by perturbing a quadratic Hamiltonian system with cubic polynomials to obtain 10 small-amplitude limit cycles bifurcating from an elementary center, for which up to 5th-order Melnikov functions are used. This demonstrates a good example in applying higher-order Melnikov functions to study bifurcation of limit cycles.

Abstract:
The problem of homoclinic bifurcations in planar continuous piecewise-linear systems with two zones is studied. This is accomplished by investigating the existence of homoclinic orbits in the systems. The systems with homoclinic orbits can be divided into two cases: the visible saddle-focus (or saddle-center) case and the case of twofold nodes with opposite stability. Necessary and sufficient conditions for the existence of homoclinic orbits are provided for further study of homoclinic bifurcations. Two kinds of homoclinic bifurcations are discussed: one is generically related to nondegenerate homoclinic orbits; the other is the discontinuity induced homoclinic bifurcations related to the boundary. The results show that at least two parameters are needed to unfold all possible homoclinc bifurcations in the systems. 1. Introduction Nonsmooth dynamical systems are naturally used to model many physical processes, such as impacting, friction, switching, and sliding systems. The study of the nonsmooth dynamical systems has attracted more and more attention in the recent decades. Piecewise-smooth systems, as an important branch of the nonsmooth dynamical systems, involve collision systems, Filippov systems, higher-order discontinuity systems, and so forth [1, 2]. In particular, the study of piecewise-linear systems is significantly important because it not only describes some processes such as circuits [3, 4], but also enables us to locally understand the bifurcation phenomena in the nonlinear systems [5]. This paper studies the planar piecewise-linear continuous vector fields with two zones. Without loss of generality, the considered plane is divided into two half-planes by the boundary coinciding with the vertical axis. The system is linear in each of the half-planes and continuous along the vertical axis. In 1998, Freire et al. [6] studied discontinuous bifurcations, proved that there exists at most one limit cycle, and proved that if the limit cycle exists then it is either attracting or repelling, which solved the problem proposed in 1991 by Lum and Chua [3]. di Bernardo et al. [1] studied discontinuity induced bifurcations, including boundary equilibrium bifurcations and grazing bifurcations of limit cycles. Concerning the limit cycle problem, Simpson and Meiss [7] investigated the Hopf branch in this kind of systems and mentioned homoclinic loops in [5]. One can refer to [8, 9] for the periodic orbit problems in piecewise-linear systems with multiple nonsmooth boundaries. Bifurcation of limit cycles by perturbing a piecewise-linear Hamiltonian system

Abstract:
本文讨论了一类含有两条切换线的平面分段光滑系统非双曲极限环的分岔. 假设该系统的未扰系统含有一个非双曲极限环且它分别与每一条切换线横截相交一次, 本文应用 Diliberto 定理和由此引出的变分引理导出了极限环的 Poincare映射, 并讨论了极限环的稳定性及其在扰动下的分岔. In this paper we consider the bifurcation of nonhyperbolic limit cycles in a class of piecewise smooth planar systems with two zones. We assume that the unperturbed system has a nonhyperbolic limit cycle that crosses every switching line transversally exactly once. By using the Diliberto theorem and variation lemma we derive the Poincare map. Then we discuss the stability and bifurcation of the limit cycle

Abstract:
In this paper, we study the problem of limit cycle bifurcation in two piecewise polynomial systems of Li\'enard type with multiple parameters. Based on the developed Melnikov function theory, we obtain the maximum number of limit cycles of these two systems.

Abstract:
In this paper, we consider some cubic near-Hamiltonian systems obtained from perturbing the symmetric cubic Hamiltonian system with two symmetric singular points by cubic polynomials. First, following Han [2012] we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center. Based on the method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be elementary center or nilpotent center. Under the condition for the singular point to be a center, we obtain the normal form of the Hamiltonian systems near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials, we consider limit cycles bifurcating from the center using the algorithm to compute the coefficients of Melnikov function. Finally, perturbing the symmetric hamiltonian system by symmetric cubic polynomials, we consider the number of limit cycles near one of the symmetric centers of the symmetric near-Hamiltonian system, which is same to that of another center.

Abstract:
We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincar\'e map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits at least two limit cycles that appear by perturbations of a period annulus. Moreover, we describe the bifurcation of the limit cycles for this class through two examples of two-parameter families of piecewise linear vector fields with three zones.

Abstract:
We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold . Treating the threshold as a bifurcation parameter that varies between 0 and , we work out a complete asymptotic and bifurcation analysis. Among other things, we show that all solutions either tend to a limit 1-cycle or to a limit 2-cycle and, we find the exact regions of attraction for these cycles depending on the size of the threshold. In particular, we show that when the threshold is either small or large, there is only one corresponding limit 1-cycle which is globally attractive. It is hoped that the results obtained here will be useful in understanding interacting network models involving piecewise constant control functions.

Abstract:
In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework. The theoretical study is illustrated with several examples.

Abstract:
An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve a quantum computer into an entanglement of states which map onto the set of all possible paths originating from a chosen vertex, and furthermore to subsequently project out all states not corresponding to Hamiltonian cycles.