where the F~ are real nonlinear functions and /~ is a real parameter. Suppose xi = 0, i= 1,. . . , n, is a solution of (1.1) for all #. Then it is well known that a point ~t = 2 at which this solution exchanges (that is, loses or gains) stability is a potential origin for a nontrivial bifurcating solution. Suppose the null solution is stable when l~ 2. Then under appropriate conditions it can be shown that if a bifurcating solution exists for/~ > 2 (supercritical) it is stable, while if it exists for/~ < 2 (subcritical) it is unstable. These are the essential bifurcation and stability results associated with the name of HOPE [1] ; they have been developed and extended by JOSEPH [2], SATTINGER [3] and others. The Hopf theory is a local theory: it refers to a sufficiently small range of values of ]/~-2] and to solutions of sufficiently small norm. Very little is known about solutions or their behaviour outside the local regime. There are several theorems which guarantee the existence of solutions in the large under certain circumstances (see, for example, RABINOWITZ [4]), and there are certain special problems which can be solved in closed form to reveal the global behaviour of solutions. One such problem, discussed in detail by PIMBLEY (pp. 4--10, [5]), demonstrates the possibility of secondary bifurcation from a supercritical branch. For real problems, however, calculation of global solutions and investigation of their stability is a very difficult matter, and has been achieved very rarely. In this paper we address ourselves to certain questions concerning the stability of steady nontrivial solutions of (1.1), whose existence in the large is assumed. We shall examine also the occurrence of secondary bifurcation* from a nontrivial solution branch of (1.1), and the relation between stability and secondary bifurcation will be discussed. Specifically, we shall treat the following questions. Given a steady nontrivial solution of (1.1), how to identify the points at which the solution exchanges stability and at which secondary bifurcation occurs? Are they necessarily the same points? What special features, if any, are held by turning points on the solu-