Bifurcation of Fredholm Maps I; The Index Bundle and Bifurcation

We associate to a parametrized family $f$ of nonlinear Fredholm maps possessing a trivial branch of zeroes an {\it index of bifurcation} $\beta(f)$ which provides an algebraic measure for the number of bifurcation points from the trivial branch. The index $\beta(f)$ is derived from the index bundle of the linearization of the family along the trivial branch by means of the generalized $J$-homomorphism. Using the Agranovich reduction and a cohomological form of the Atiyah-Singer family index theorem, due to Fedosov, we compute the bifurcation index of a multiparameter family of nonlinear elliptic boundary value problems from the principal symbol of the linearization along the trivial branch. In this way we obtain criteria for bifurcation of solutions of nonlinear elliptic equations which cannot be achieved using the classical Lyapunov-Schmidt method.Comment: 42 pages. Changes: added Lemma 2.31 and a reference + minor corrections. To appear on TMN

Bifurcation of Fredholm maps II; The dimension of the set of bifurcation points

We obtain an estimate for the covering dimension of the set of bifurcation points for solutions of nonlinear elliptic boundary value problems from the principal symbol of the linearization of the problem along the trivial branch of solutions.Comment: 15 pages, corrected typos, minor changes; La Matematica e le sue Applicazioni N5(2010). To appear on TMN

Parity and generalized multiplicity

Assuming that X and Y are Banach spaces and that T is a path of linear Fredholm operators with invertible endpoints, in [F-Pl] we defined a homotopy invariant "the parity of T . The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove that, generically, the parity is a mod 2 count of the number of transversal intersections of T with the set of singular operators, that at an isolated singular point of x of T the local parity remains invariant under Lyapunov-Schmidt reduction, and that it coincides with the mod 2 reduction of any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data

Orientability of Fredholm families and topological degree

We construct a degree theory for oriented Fredholm mappings of index zero between open subsets of Banach spaces and between Banach manifolds. Our approach is based on the orientation of Fredholm mappings: it does not use Fredholm structures on the domain and target spaces. We provide a computable formula for the change in degree through an admissible homotopy that is necessary for applications to global bifurcation. The notion of orientation enables us to establish rather precise relationships between our degree and many other degree theories for particular classes of Fredholm maps, including the Elworthy-Tromba degree, which have appeared in the literature in a seemingly unrelated manner

Complementing maps, continuation and global bifurcation

We state, and indicate some of the consequences of, a theorem whose sole assumption is the nonvanishing of the Leray- Schauder degree of a compact vector field, and whose conclusions yield multidimensional existence, continuation and bifurcation result

On the covering dimension of the set of solutions of some nonlinear equations

We prove an abstract theorem whose sole hypothesis is that the degree of a certain map is nonzero and whose parametric equations are studied using cohomological mconclusions imply sharp, multidimensional continuation results. Applications are given to nonlinear partial differential equations

Topological invariants of bifurcation

I will shortly discuss an approach to bifurcation theory based on elliptic topology. The main goal is a construction of an index of bifurcation points for $C^1$-families of Fredholm maps derived from the index bundle of the family of linearizations along the trivial branch. As illustration, I will present an application to bifurcation of homoclinic solutions of non-autonomous differential equations from a branch of stationary solutions

Global bifurcation of homoclinic trajectories of discrete dynamical systems

We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving the topological properties of the asymptotic stable bundles