Theorems, Bendixon

It may be demonstrated (the Poincare-Bendixon theorem, see Appendix A2) that for the autonomous system in two variables (5.2) a limit set can only be a point or a closed curve. Linear systems, gradient and Hamiltonian systems, will be shown in Appendix A1 to be uncapable of having a limit cycle. 

On the other hand, it follows from the Poincare-Bendixon theorem (see Appendix A2.2) that an attractor cannot have a dimension equal to two (plane), one (line) or zero (point), since a chaotic nature of the limit set is then impossible (only stationary points and limit sets are then possible). Hence, a conclusion follows, confirmed by other methods, that the attractor of the system of equations (5.14) has a fractional dimension (more exactly, 2 + D, where D is a small positive number), see Appendix A2.7. 

It should be stressed that the Poincare-Bendixon theorem given above does not hold for systems of three or more autonomous equations hence, in such systems non-periodicity of a nonstationary trajectory remaining within a confined region is possible. 

As we have concluded in Chapter 5, in the case of the Lorenz system a trajectory always remains within a confined region of the phase space, being non-periodical. For t - oo, the trajectory approaches a certain limit set the Lorenz attractor. It follows from the Liouville theorem that the Lorenz attractor has a zero volume (since divF < 0). This implies that, apparently, the Lorenz attractor is a point (dimension zero), a line (dimension one), or a plane (dimension two). Then, however, the trajectory for t -> oo would have remained within a confined region on the plane and, by virtue of the Poincare-Bendixon theorem. Hence, a conclusion follows that the Lorenz attractor has a fractional dimension, larger than two. 

This means that in the domain Q) consisting of 3) but without the origin P (0, 0), there must exist a limit cycle as solution of (4.53) according to the Poincare-Bendixon theorem [4.2,12]. The solution computed and shown in Fig. 4.8 a (to be explained later) verifies this statement. 

Theorem-. The domain % of the d-x plane bounded internally by an infinitesimally small ellipse enclosing the origin and externally by the straight lines d = do, X = 1, d = -do and x = - 1 (Fig. 5.3) fulfills the premise of the Poin-care-Bendixon theorem that ... 

Conclusion From the Poincare-Bendixon theorem it now follows that within Sc a limit cycle, i.e. a periodic solution of the equations of motion C (t), must exist and that all other solutions starting from any point within Sc approach this limit cycle. Sc is a domain of attraction for C (t). 

Bendixon, negative criterion of, 333 Bendixon-Poincare Theorem, 333 Berezetski, V. B., 723... 

The theorems of Bendixon and Dulac are generalized for the case of n-dimensional systems with first integrals. It is shown as an illustration that the classical Michaelis-Menten reaction has no oscillatory solution. As another illustration we show that reactions with M components composed of M-1 atoms have no oscillatory solution either C6j. 

J. Toth "Bendixon Type Theorems with Applications", in Coll. Math. Soc. J.Bolyai /in preparation/. 

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