Abstract. This paper contains a description of a program designed to find all the solutions of systems of two real equations in two real unknowns which uses detailed information about the critical set of the associated function from the plane to the plane. It turns out that the critical set and its image are highly structured, and this is employed in their numerical computation. The conceptual background and details of implementation are presented. The most important features of the program are the ability to provide global information about the function and the robustness derived from such topological information.
We consider the operator F (u) = u ′ + f (t, u(t)) acting on periodic real valued functions. Generically, critical points of F are infinite dimensional Morin-like singularities and we provide operational characterizations of the singularities of different orders. A global Lyapunov-Schmidt decomposition of F converts F into adapted coordinates, F(ṽ,ū) = (ṽ,v), whereṽ is a function of average zero and bothū andv are numbers. Thus, global geometric aspects of F reduce to the study of a family of one-dimensional maps: we use this approach to obtain normal forms for several nonlinearities f . For example, we characterize autonomous nonlinearities giving rise to global folds and, in general, we show that F is a global fold if all critical points are folds. Also, f (t, x) = x 3 − x, or, more generally, the Cafagna-Donati nonlinearity, yield global cusps; for F interpreted as a map between appropriate Hilbert spaces, the requested changes of variable to bring F to normal form can be taken to be diffeomorphisms. A key ingredient in the argument is the contractibility of both the critical set and the set of non-folds for a generic autonomous nonlinearity. We also obtain a numerical example of a polynomial f of degree 4 for which F contains butterflies (Morin singularities of order 4)-it then follows that F (u) = v has six solutions for some v.1991 Mathematics Subject Classification. Primary 58C27, 34B15, 34L30; Secondary 47H15.
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