Yirong Liu

Stability and bifurcations of limit cycles of the equator in a class of cubic polynomial systems

Abstract In this paper, we study the appearance of limit cycles from the equator in a class of cubic polynomial vector fields with no singular points at infinity and the stability of the equator of the systems. We start by deducing the recursion formula for quantities at infinity in these systems, then specialize to a particular case of these cubic systems for which we study the bifurcation of limit cycles from the equator. We compute the quantities at infinity with computer algebraic system Mathematica 2.2 and reach with relative ease an expression of the first six quantities at infinity of the system, and give a cubic system, which allows the appearance of six limit cycles in the neighborhood of the equator. As far as we know, this is the first time that an example of cubic system with six limit cycles bifurcating from the equator is given. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of quantities at infinity is linear and then avoids complex integrating operations. Therefore, the calculation can be readily done with using computer symbol operation system such as Mathematica.

A quintic polynomial differential system with eleven limit cycles at the infinity

In this article, a recursion formula for computing the singular point quantities of the infinity in a class of quintic polynomial systems is given. The first eleven singular point quantities are computed with the computer algebra system Mathematica. The conditions for the infinity to be a center are derived as well. Finally, a system that allows the appearance of eleven limit cycles in a small enough neighborhood of the infinity is constructed.

The bifurcation of limit cycles in Zn-equivariant vector fields

Abstract In this paper, we study the bifurcation of limit cycles from fine focus in Z n -equivariant vector fields. An approach for investigating bifurcation was obtained. In order to show our work is efficacious, an example on bifurcations behavior is given, namely five order singular points values are given in the seventh degree Z 8 -equivariant systems. We discuss their bifurcation behavior of limit cycles, and show that there are eight fine focuses of five order and five small amplitude limit cycles can bifurcate from each. So 40 small amplitude limit cycles can bifurcate from eight fine focuses under a certain condition. In terms of the number of limit cycles for seventh degree Z 8 -equivariant systems, our results are good and interesting.

A class of ninth degree system with four isochronous centers

In this paper we study a class of ninth degree system and obtain the conditions that its four singular points can be general centers and isochronous centers (or linearizable centers) at the same time by computing carefully and strict proof. What is worth mentioning is that the expressions of Liapunov constants and periodic constants are simpler, and recursive formulas of node point values are given for the first time, which is a new effective criterion for verifying isochronous centers.

General center conditions and bifurcation of limit cycles for a quasi-symmetric seventh degree system

In this paper, we study a class of quasi-symmetric seventh degree system. By making two appropriate transformations of system (3) and calculating general focal values carefully, we obtain the conditions that the infinity and the elementary focus (-12,0) become centers at the same time. Moreover, 12 limit cycles including 6 small limit cycles from the elementary focus and 6 large limit cycles from the infinity can occur under a certain condition. What is worth mentioning is that similar conclusions are less and our work is new in terms of research about quasi-symmetric systems up till now.

Global exponential periodicity for BAM neural network with periodic coefficients and continuously distributed delays

By constructing a suitable Lyapunov function and using some analysis techniques, rather than employing the continuation theorem of coincidence degree theory as in other literature, a sufficient criterion is obtained to ensure the existence and global exponential stability of periodic solution for the bidirectional associative memory neural network with periodic coefficients and continuously distributed delays. The obtained result is less restrictive to the BAM neural network than the previously known criteria. And it can be applied to the BAM neural network in which signal transfer functions are neither bounded nor differentiable. In addition, an example and its numerical simulation are given to illustrate the effectiveness of the obtained result.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

Center problem and bifurcation behavior for a class of quasi analytic systems

Abstract Results about the study of nonanalytic systems’ center-focus and bifurcations of limit cycles are hardly seen in published references up till now. In this paper, we investigated the problems of determining center or focus and bifurcations for a class of planar quasi cubic analytic systems. The recursive formula to figure out generalized focal values is given, ulteriorly the conditions for four limit cycles from the origin or the point at infinity are obtained and center problems are considered. What is worth pointing out is that we offer a kind of interesting phenomenon that the exponent parameter λ control the non-analyticity of studied system (3.8) in this paper. In terms of nonanalytic differential systems, our work is new.

Limit Cycle Bifurcation of Infinity and Degenerate Singular Point in Three-Dimensional Vector Field

Our work focuses on investigating limit cycle bifurcation for infinity and a degenerate singular point of a fifth degree system in three-dimensional vector field. By using singular value method to compute focal values carefully, we give the expressions of the focal values (Lyapunov constants) at the origin and at infinity. Moreover, we obtain that four limit cycles at most can bifurcate from the origin and three limit cycles can bifurcate from infinity. At the same time, we show the structure of limit cycles from the origin and the infinity. It is interesting for this kind of nonlinear phenomenon that a string of large limit cycles encircle a string of small limit cycles by simultaneous Hopf bifurcation, which is hardly seen for similar published results in three-dimensional vector field, our result is new.