We prove several new comparison results and develop the monotone iterative technique to show the existence of extremal solutions to a kind of periodic boundary value problem (PBVP) for nonlinear integro-differential equation of mixed type on time scales.
In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z3 equivariant system can have 13 limit cycles.
In this article, we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop. We obtain an explicit formula to compute the first four coeffcients, and then identify the method of finding at least 7 limit cycles near the double homoclinic loop using these coefficients. Finally, we present some interesting applications.
Yang Junmin Han Maoan (Dept.of Math.,Shanghai Normal University,Shanghai 200234)
This paper concerns the number and distributions of limit cycles in a Z 2-equivariant quintic planar vector field. 25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation. It can be concluded that H(5) ? 25 = 52, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to study the weakened 16th Hilbert problem.
In this paper, we are concerned with a cubic near-Hamiltonian system, whose unperturbed system is quadratic and has a symmetric homoclinic loop. By using the method developed in [12], we find that the system can have 4 limit cycles with 3 of them being near the homoclinic loop. Further, we give a condition under which there exist 4 limit cycles.
