Invariant manifolds for some autonomous systems

Abstract

Under certain restrictions we show that an invariant (n - 1)-manifold in the x-space bifurcates (branches off) from the fixed periodic solutions for values of iu near ,to. This process will be known as bifurcation from a periodic solution. Poincare [10] has shown that in the plane such behavior may occur. Various other authors have studied manifolds defined by solutions of ordinary differential equations. However, most of this work is concerned with the perturbation problem. For an extensive bibliography on this topic see W. T. Kyner [6]. The basic theorem of N. Levinson [9] established the existence of an invariant manifold based on x = -(t) in the extended phase space. This manifold was topologically a torus, that is, Hln S I = S I x S' x ... x S I (Sn is the unit n-sphere) in the (x,t)-space. It was assumed that the real parts of n-1 of the characteristic exponents based on +(t) were negative for all values of the parameter iu. This theorem has been generalized by various authors including Hufford [4], Kyner [5; 7], Diliberto and Hufford [3], and Diliberto [1; 2]. The assumption of stability (the real parts of n - 1 of the characteristic exponents are negative) has been replaced by the assumption of conditional stability (the real parts of n - 1 of the characteristic exponents are nonzero). These results differ from ours in three respects. The first is that our invariant manifold lies in the x-space;