Poincare-Bendixson theory

Theorem C.7 bears a strong resemblance to the Poincare-Bendixson theorem stated in Chapter 1. It will be used in Chapter 4 for the case where (C.l) is a competitive system, that is, for a system (C.l) where -/ is cooperative. Note that the omega (alpha) limit set of a competitive system is the alpha (omega) limit set of the time-reversed cooperative system, so Theorems C.5, C.6, and C.7 apply to competitive systems. Unlike cooperative systems, competitive systems can have attracting periodic orbits. For more on the Poincare-Bendixson theory of competitive and cooperative systems in see [S3], [SWl], and [ZS]. 

The proof of sufficiency of the conditions of the Andronov-Pontryagin theorem relies heavily on the Poincare-Bendixson theory which gives a classification of every possible type of trajectories in two-dimensional systems on the plane (see Sec. 1.3). We refer the reader to the books [11, 12] for further details. 

The Poincare-Bendixson theory is also applicable for systems on a cylinder, as well as on a two-dimensional sphere. As for other compact surfaces like tori, pretzels (spheres with a handle) etc., there may exist vector fields that possess, besides equilibria and limit cycles, unclosed Poisson-stable trajectories as well. 

If the inequality is reversed then the rest point E. is unstable - a repeller. The Poincar -Bendixson theorem then allows one to conclude that there exists a limit cycle. Unfortunately, there may (theoretically) be several limit cycles. If all limit cycles are hyperbolic then there is at least one asymptotically stable one, for if there are multiple limit cycles the innermost one must be asymptotically stable. Moreover, since all trajectories eventually lie in a compact set, there are only a finite number of limit cycles and the outermost one must be asymptotically stable. Since the system is (real) analytic, one could also appeal to results for such systems. For example, Erie, Mayer, and Plesser [EMP] and Zhu and Smith [ZSJ show that if E is unstable then there exists at least one limit cycle that is asymptotically stable. Stability of limit cycles will be discussed in the next section. We make a brief digression to outline the principal parts of this theory, and then return to the food-chain problem. 

How can oscillations, in particular chemical oscillations, be explained within the mathematical dynamic theory Unfortunately, there is still no rigorous theory for distinguishing multidimensional models of self-sustained oscillations. A typical strategy is first finding the models and parametric domains in which these oscillations do not exist. For instance, according to the so-called Poincare-Bendixson criterion (which is only valid for systems with two variables), if the sum... 

We have already noticed that the theory of structurally stable systems of second order on a plane is based essentially on the theory of Poincar Bendixson, and on the classification of all possible kinds of motions. Below is the diagram suggested by Andronov which describes the general classification of motions due to Birkhoff. 

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