The Index Change and Global Bifurcation for Flows with a First Integral

The existence, continuation and bifurcation of solutions of fixed period τ of one-parameter families of ordinary differential equations with a first integral may be studied using the topological degrees defined in [6] and [7] and the change of index formula given in [8]. The purpose here is threefold. First, suppose that 〈f(x), ∇V(x)〉 = 0, so that V is a first integral for x = f(x). We examine how the behaviour of f restricts that of V and vice versa, and how the periodic spectra of linear ordinary differential equations are affected by the presence of a linear or non-linear (which occurs surprisingly naturally) first integral. Second, we give classes of one-parameter families of ordinary differential equations where the change of index in [8] is non-zero. These results depend on the Jacobian of the first integral at an equilibrium being non-singular. Third, we derive global bifurcation theorems for τ-periodic solutions (τ-fixed) when the Jacobian of the first integral is non-singular at equilibrium. Then, to treat significant classes of problems where the Jacobian is singular, we develop a new formula which mimics the role of the change of index formula in non-degenerate problems. This itself is not a change of index because, in such degenerate cases, the index change may not be well defined. The evaluation of this formula involves consideration of terms of order higher than quadratic in the Taylor expansion of the first integral. When it is non-zero, we obtain global bifurcation results (which are new even when regarded as local theorems) for one-parameter families of ordinary differential equations which have a singular first integral.