By Terence Tao

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This publication originated from an introductory lecture direction on dynamical structures given through the writer for complex scholars in arithmetic and physics at ETH Zurich. the 1st half facilities round volatile and chaotic phenomena as a result of the incidence of homoclinic issues. The lifestyles of homoclinic issues complicates the orbit constitution significantly and offers upward thrust to invariant hyperbolic units within reach.

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Let P(x, y) and Q(x, y) have second-order continuous partial derivatives in a simply connected region G, and assume that P(x, y) = 0 can be represented by y = f(x), and Q(x, y) = 0 can be represented by x = cp(y). Let the functions a ( 1 aQ) A(x,y) =ax Q ay I a (1 aP) B(x,y) = - ay Pax have the same signinG, never change sign, and assume that in any subregion at least one of A, B is not identically zero. Also assume that near the upper (lower) half of y = f(x), Band p-tapjax have opposite (same) sign, and near the right (left) side of x = cp(y), A and Q- 1 aQjay have opposite (same) sign.

21) is also a necessary condition for being stable (unstable) or semistable. 3. 21) is called a k- fold limit cycle. From this definition it is easy to see if r is a k-fold limit cycle, then no = 0 is a k-multiple root of the equation 'IJ(n0 ) = 0. If we draw the linen= no and the curve n = n(l, no) in the (n 0 , n) plane, then the origin is a k-fold point of intersection of these two curves. e. the conditions w'(O) < 0 (> 0) and fa' H(s) ds < 0 (> 0) are equivalent. 17) with respect to no, and obtain w'(no) In order to compute equation =fa' F~(s,n(s,no))n~0 (s,no)ds.

15, we only use the existence part of this theorem. 15. §3. ROTATED VECTOR FIELDS 47 trajectories of r a, while meeting the annular region, cross it in the exterior-tointerior direction. 15 (see the footnote to the preceding paragraph) we know that in this annular region there will again exist a new closed trajectory of F(a) which should lie in the interior of r a· This contradicts the definition of r a· Similarly, inN£, F(a) must have an outermost closed trajectory r a', which must be an externally stable cycle.