By L. Smith

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**Additional resources for Algebraic Topology, Gottingen 1984**

**Example text**

Then at time t2 the curve exits the region U1 . Therefore the ordered pair of vectors, consisting of the tangent vector to the curve and the vector of the dynamical system, at the times t1 and t2 determine opposite orientations of the plane. Consequently, at some time in between, the tangent vector and the vector of the dynamical system are collinear. REMARx. The notion of contact point of a curve and a dynamical system was introduced by Poincar6 in his memoir 143]. , Part I, p. 377) has contact points with the dynamical system.

The points a1 and b1 can be connected by a curve y lying in r, since r is connected. Let a2 lie over a1 on the boundary of the tubular neighbourhood of r on the same side as the curve segment that connects a with a1 . Lift y to the boundary of the tubular neighbourhood of F so that the lifted curve starts at a2 and projects to y. The end of the lifted curve lies over b1 . It can lie either on the same side of r as the curve segment connecting b to b1 , or on the opposite side. In the first case we can close the curve connecting a and b without crossing r, in the second case by crossing the submanifold exactly once, and that crossing is transversal.

Any connected component of a separating solution is either a cycle or a noncompact trajectory that goes to infinity. EXAMPLE 3. A nonsingular level curve of the function H(xi , x2) = C, oriented as the boundary of the region H(xl , x,) > c, is a separating solution 11 11. TWO SIMPLE VERSIONS OF THE THEORY OF FEWNOMIALS 22 of the Hamiltonian system r1 = 8H/8x ace = -eH/8x1 . DEFINITION. A contact point of a curve and a dynamical system in the plane is a point of the curve in which the tangent vector to the curve and the vector of the dynamical system are collinear.