Bifurcation from a Simple Eigenvalue

Before giving a proof, note that the components of are readily obtainable Xc-Xi and y,. is the root of a quadratic. The condition is stated in the form (3.4) to avoid the complicated expression that would result from using the quadratic formula. 

Proof of Theorem 5.3. Condition (3.4) makes E locally asymptotically stable. By the Poincare-Bendixson theorem, it is necessary only to show that with condition (3.4) there are no limit cycles. Suppose there were a limit cycle. However, there is at most a finite number of limit cycles and each must contain in its interior. Hence there is a periodic trajectory V that contains no other periodic trajectory in its interior. Intuitively speaking, r is the trajectory closest to the rest point. The constant term in the formula given in Lemma 5.1 is negative. The corollary shows that P is asymptotically stable. This is a contradiction, since the rest point is asymptotically stable - that is, between the two there must be an unstable periodic orbit.  

When the inequality in (3.4) is reversed, there will be a periodic orbit (by an application of the Poincare-Bendixson theorem). By our assumption of hyperbolicity, this orbit must be asymptotically orbitally stable since it is so from the inside. These comments establish the next result. 

Kuang [Kl] has shown that if the parameters are such that 

In Section 4 it was shown that a food chain depending on the nutrient S, a first-level consumer x, and a predator could possess a periodic solution. From the standpoint of the full system (2.2) or the simplified system 

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The new mathematics that is introduced here is elementary bifurcation theory, in particular, bifurcation from a simple eigenvalue. Although the necessary theorems will not be proved, the material will be discussed in some detail. 

Although stability may in principle be computed, the calculation is extremely complicated. Numerical calculations suggest the asymptotic stability of the limit cycle, but the stability has not been rigorously established. Assuming that the solution is asymptotically stable, a secondary bifurcation can be shown to occur. The argument is quite technical and requires a form of a Poincare map in the appropriate function space it is analogous to the bifurcation theorem used in Chapter 3 for bifurcation from a simple eigenvalue. The principal theorem takes the form of a bifurcation statement. 

The situation which we consider here is a particular case of Theorem 13.9 of the next section. It follows from this theorem (applied to the system in the reversed time) that a single saddle periodic orbit L is born from a homoclinic loop it has an m-dimensional stable manifold and a two-dimensional unstable manifold. This result is similar to Theorem 13.6. Note, however, that in the case of a negative saddle value the main result (the birth of a unique stable limit cycle) holds without any additional non-degeneracy requirements (the leading stable eigenvalue Ai is nowhere required to be simple or real). On the contrary, when the saddle value is positive, a violation of the non-degeneracy assumptions (1) and (2) leads to more bifurcations. We will study this problem in Sec. 13.6. 

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