By L. Garrido

ISBN-10: 3540122761

ISBN-13: 9783540122760

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**Sample text**

4, which is bounded but does not correspond with a periodic solution, the w-limitset w(-y+) contains a critical point or it consists of a closed orbit. In the proof the following topological property is essential: a sectionally smooth, closed orbit e in R2, a cycle, separates the plane into two parts S. and S;. The sets S. and S; are disjunct and open, e is the boundary of both S. and S;; we can write R2 Ie = s. U S; (Jordan). Two simple tools will be useful. 4; such a point we shall call an ordinary point.

Does the result agree with the Hamiltonian nature of the problem? d. Sketch the phase-plane for various values of >.. 2-5. Find the critical points of the system Characterise the critical points by linear analysis and determine their attraction properties. 2-6. Consider the system x=x iJ = Y· 26 a. Find a first integral. h. Can we derive the equations from a Hamilton function? 2-7. f(x) = o. So the phase-flow is volume-preserving. Does this mean that a first integral exists; solve this question for n = 2.

The sets S. and S; are disjunct and open, e is the boundary of both S. and S;; we can write R2 Ie = s. U S; (Jordan). Two simple tools will be useful. 4; such a point we shall call an ordinary point. Secondly the concept of transversal. 6 There exist several, related proofs for the Poincare-Bendixson theorem; see for instance Coddington and Levinson (1955). Here we are using basically the formulation given by Hale (1969). We shall construct the proof out of several lemmas. 6 Consider the transversal 1 which is supposed to be a closed set, the interior of 1 is lo.

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Categories: Differential Equations