As we know, the second part of the Hilbert problem is to find the maximal number and relative locations of limit cycles of polynomial systems of degree n. Let H(n) denote this number, which is called the Hilbert number. Then the problem of finding H(n) is divided into two parts: find an upper and lower bounds of it. For the upper bound there are important works of Écalle [1990] and IIyashenko a...

In the studies of nonlinear dynamics, phase plan plot is a most commonly used tool for solution characteristics interpretation. Phase plan plot provides an adequate representation of the dynamic characteristics of single solution, but it does not provide information on the interrelation between neighboring solutions; therefore, the evolution of solutions is studied by examining fragmented infor...

The main purpose of this paper is to give a reasonably comprehensive discussion of what is commonly referred to as the bifurcation analysis applied to an indirect field oriented control of induction machines (IFOC). In the current work, we study the appearance of self-sustained oscillations in AC drives and compute their corresponding stability margins. As the dynamics is explored, a transition...

The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particu...

Novak and Tyson have proposed a realistic mathematical model of the biochemical mechanism that regulates M-phase promoting factor (MPF), the major enzymatic activity controlling mitotic cycles in frog eggs, early embryos, and cell-free egg extracts. We use bifurcation theory and numerical methods (AUTO) to characterize the codimension-one and -two bifurcation sets in this model. Our primary bif...

BIFURCATION SET AND LIMIT CYCLES FORMING COMPOUND EYES IN A PERTURBED HAMILTONIAN SYSTEM JIBIN LI AND ZHENRONG LIU In this paper we consider a class of perturbation of a Hamiltonian cubic system with 9 finite critical points . Using detection functions, we present explicit formulas for the global and local bifurcations of the flow . We exhibit various patterns of compound eyes of limit cycles ....

Let X be a homogeneous polynomial vector field of degree 2 on S. We show that if X has at least a non–hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on S is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 Hilbert’s ...

We study the bifurcation of limit cycles from the periodic orbits of a 4–dimensional center in a class of piecewise linear differential systems, which appears in a natural way in control theory. Our main result shows that three is an upper bound for the number of limit cycles, up to first order expansion of the displacement function with respect to the small parameter. Moreover, this upper boun...

In this paper, we consider the weakened Hilbert’s 16th problem for symmetric planar perturbed polynomial Hamiltonian systems. In particular, a perturbed Hamiltonian polynomial vector field of degree 9 is studied, and an example of Z10-equivariant planar perturbed Hamiltonian systems is constructed. With maximal number of closed orbits, it gives rise to different configurations of limit cycles. ...

In recent papers we have introduced a method for the study of limit cycles of the Liénard system : ẋ = y − F (x) , ẏ = −x , where F (x) is an odd polynomial. The method gives a sequence of polynomials Rn(x), whose roots are related to the number and location of the limit cycles, and a sequence of algebraic approximations to the bifurcation set of the system. In this paper, we present a variant ...