Limit Cycles Bifurcations for a Class of Kolmogorov Model in Symmetrical Vector Field

The problem of limit cycles for Kolmogorov model is interesting and significant both in theory and applications. Our work is concerned with limit cycles bifurcations problem for a class of quartic Kolmogorov model with two positive singular points (i.e. (1, 2) and (2, 1)). The investigated model is symmetrical with regard to y = x. We show that each one of points (1, 2) and (2, 1) can bifurcate five small limit cycles at the same step under a certain condition. Hence, the two positive singular points can bifurcate ten limit cycles in sum, in which six cycles can be stable. In terms of symmetrical Kolmogorov model, published references are less. In terms of the Hilbert Number of Kolmogorov model, our results are new.

ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES IN A QUINTIC PLANAR VECTOR FIELD

This paper is concerned with the number and distributions of limit cycles in a Z2-equivariant quintic planar vector field. By applying qualitative analysis method of differential equation, we find that 28 limit cycles with four different configurations appear in this special planar polynomial system. It is concluded that H(5) ≥ 28 = 52+ 3, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to the study of the second part of 16th Hilbert problem.

BIFURCATION OF LIMIT CYCLES IN A FOURTH-ORDER NEAR-HAMILTONIAN SYSTEM

This paper is concerned with bifurcation of limit cycles in a fourth-order near-Hamiltonian system with quartic perturbations. By bifurcation theory, proper perturbations are given to show that the system may have 20, 21 or 23 limit cycles with different distributions. This shows that H(4) ≥ 20, where H(n) is the Hilbert number for the second part of Hilbert's 16th problem. It is well known that H(2) ≥ 4, and it has been recently proved that H(3) ≥ 12. The number of limit cycles obtained in this paper greatly improves the best existing result, H(4) ≥ 15, for fourth-degree polynomial planar systems.

HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS

The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert's 16th problem? Section 2: The first part of Hilbert's 16th problem. Section 3: The second part of Hilbert's 16th problem: introduction. Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop. Section 5: Finiteness problem. Section 6: The weakened Hilbert's 16th problem. Section 7: Global and local bifurcations of Zq–equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n.

BIFURCATIONS OF LIMIT CYCLES IN A Z3-EQUIVARIANT VECTOR FIELD OF DEGREE 5

A concrete numerical example of Z3-equivariant planar perturbed Hamiltonian vector field of fifth degree having 23 limit cycles and a configuration of compound eyes are given, by using the bifurcation theory of planar dynamical systems and the method of detection functions. It gives rise to the conclusion: the Hilbert number H(5) ≥ 23 for the second part of Hilbert's 16th problem.