Existence And Bifurcation Of One-parameter Periodic Orbits Of A Class Of3-dimensional Differential Systems

In this paper, we mainly study the conditions of the existence and bifurcation of periodic orbit of one class of3-dimensional differential systems. Based on the above results, the paper studies the existence and the shape of the manifolds formed by periodic orbits in R3and so on.We mainly use two kinds of research method:the first method:we first transform the3-dimensional differential system into one-dimensional periodic system. Further we make a time rescaling to the one-dimensional periodic system. We transform it into a2π-periodic system. Applying the average, we obtain the sufficient conditions of the global existence and bifurcation of one-parameter periodic orbits. Further we obtain the sufficient conditions of existence of invariant surfaces consisting of periodic orbits.the second method:we first make the cylinder coordinate transformation to the3-dimensional differential system and obtain a one-dimensional27r-periodic system. Ap-plying some lemmas, we obtain the sufficient conditions of the local existence and bi-furcation of one-parameter periodic orbits. Further we obtain the sufficient conditions of existence of invariant surfaces consisting of periodic orbits.Applying the general results to some special polynomial systems, we obtain some invariant spheres and torus.