This paper intends to explore bifurcation behavior of limit cycles for a cubic Hamiltonian system with quintic perturbed terms using both qualitative analysis and numerical exploration. To obtain the maximum number of limit cycles, a quintic perturbed function with the form of R(x, y, λ )= S(x, y, λ )= mx 2 + ny 2 + ky 4 − λ is added to a cubic Hamiltonian system, where m, n, k and λ are all variable. The investigation is based on detection functions which are particularly effective for the perturbed cubic Hamiltonian system. The study reveals that, for the Hamiltonian system [equation (1.5) in the introduction] with the perturbed terms mentioned above, there are 15 limit cycles if 15.1149 <λ< 15.1249; and 11 limit cycles if 15.1102 <λ< 15.1149. As numerical illustration, we numerically predict the detection curves and display graphically the distribution of limit cycles for the proposed perturbed Hamiltonian system.

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Limit cycle analysis on a cubic Hamiltonian system with quintic perturbed terms

Bifurcation at the equator for a class of quintic polynomial differential system

The quasi-homogeneous polynomial differential systems have been studied from many different points of view. In this paper, center problem and bifurcation of limit cycles for p : q homogeneous weight systems are studied. Some properties of successive function and focus values are discussed. Furthermore, the method of computing focal values is given. As an example, center problem and bifurcation of limit cycles for 2:3 homogeneous weight system are studied, three or five limit cycles in the neighborhood of origin can be obtained by different perturbations.

Bifurcation of limit cycles and center problem for p:q homogeneous weight systems

Bifurcation of limit cycles and pseudo-isochronous center at degenerate singular point in a septic system

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.

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

The center problem and bifurcation of limit cycles for degenerate singular points are far to be solved in general. In this paper, we study center conditions and bifurcation of limit cycles at the degenerate singular point in a class of quintic polynomial vector field with a small parameter and eight normal parameters. We deduce a recursion formula for singular point quantities at the degenerate singular points in this system and reach with relative ease an expression of the first five quantities at the degenerate singular point. The center conditions for the degenerate singular point of this system are derived. Consequently, we construct a quintic system, which can bifurcates 5 limit cycles in the neighborhood of the degenerate singular point. The positions of these limit cycles can be pointed out exactly without constructing Poincare cycle fields. 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 singular point quantities at degenerate singular point is linear and then avoids complex integrating operations.

Haibo Chen

Papers

Center conditions and bifurcation of limit cycles at degenerate singular points in a quintic polynomial differential system