CHAOTIC MOTION GENERATED BY DELAYED NEGATIVE FEEDBACK .2. CONSTRUCTION OF NONLINEARITIES

We construct a smooth function g :IR --> IR with xi g*(xi) < 0 for xi not equal 0 such that the equation (g> x'(t) = g*(x(t - 1)) has a sl owly oscillating periodic solution y, and a slowly oscillating solutio n z whose phase curve is homoclinic with respect to the orbit o of y in the space C = C-0([-1, 0], R). For an associated Poincare map we ob tain a transversal homoclinic loop. The proof of transversality employ s a criterion which uses oscillation properties of solutions of variat ional equations. The main result is that the trajectories (psi(n))(-in finity)(infinity) of the Poincare map in a neighbourhood of the homocl inic loop form a hyperbolic set on which the motion is chaotic.