Homoclinic bifurcations in four-dimensional vector fields are investigated by setting up a local coordinate near a homoclinic orbit. This homoclinic orbit is principal but its stable and unstable foliations take inclination flip. The existence, nonexistence, and uniqueness of the 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold 1-periodic orbit and three-fold 1 -periodic orbit are also obtained. It is indicated that the number of periodic orbits bifurcated from this kind of homoclinic orbits depends heavily on the strength of the inclination flip.
SHUI Shuliang & ZHU Deming College of Mathematics and Physics, Zhejiang Normal University, Jinhua 321004, China
A persistence theorem for resonant invariant tori with non-Hamiltonian perturbation is proved. The method is a combination of the theory of normally hyperbolic invariant manifolds and an appropriate continuation method. The results obtained are extensions of Chicone's for the three dimensional non-Hamiltonian systems.
By using the continuation theorem of Mawhin's coincidence degree theory, a sufficient condition is derived for the existence of positive periodic solutions for a distributed delay competition model {u'(t)=u(t)[r1(t)-a1(t)u(t)-b1(t)∫-T^1 L1(s)u(t+s)ds-c1(t)∫-T^0(K1(s)v(t+s)ds],v'(t)=v(t)[r2(t)-a2(t)v(t)-b2(t)∫-T^0L2(s)v(t+s)ds-c2(t)∫-t^0 K2(s)u(t+s)ds],where r1 and r2 are contiuous w-periodic functions in R+=[0,∝] with ∫0^w ri(t)dt>0,ai,ci(i=1,2)are positive continuous w-periodic functions in R+=[0,∞），bi(i=1,2)is nonnegative continuous w-periodic function in R+=[0,∞),w and T are positive constants,Ki,Li∑C（[-T，0]，（0，∞）and ∫-T^0 Ki(s)ds=1,∫-T^0 Li(s)ds=1,i=1,2.some known results are improved and extended.
Xian-yi Li, De-ming ZhuDepartment of Mathematics, East China Normal University, Shanghai 200062, China Department of Mathematics and Physics, Nanhua University, Hengyang 421001, China
In this paper, the qualitative behavior of solutions of the bobwhite quail pop-ulation modelwhere 0 ＜ a ＜ 1 ＜ a + b,p,c ∈ (0, ∞) and k is a nonnegative integer, is investigated.Some necessary and suficient as well as sufficient conditions for all solutions of the modelto oscillate and some sufficient conditions for all positive solutions of the model to benonoscillatory and the convergence of nonoscillatory solutions are derived. Furthermore,the permanence of every positive solution of the model is also showed. Many known resultsare improved and extended and some new results are obtained for G. Ladas' open problems.
