1,721,020 research outputs found
Topological invariants of bifurcation
Get PDFI 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 -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
Bifurcation of Homoclinics of Hamiltonian Systems
No full textWe obtain sufficient conditions for bifurcation of homoclinic trajectories of nonautonomous Hamiltonian vector fields parametrized by a circle, together with estimates for the number of bifurcation points in terms of the Maslov index of the asymptotic stable and unstable bundles of the linearization at the stationary branc
Bifurcation of homoclinics
No full textWe show that homoclinic trajectories of nonautonomous vector fields parametrized by a circle bifurcate from the stationary solution when the asymptotic stable bundles of the linearization at plus and minus infinity are "twisted" in different way
The homotopy theory of weighted mappings
No full textWe introduce the basic properties of the homotopy theory of weighted maps and show that the hom functor in this category is a representable functo
Relation between the homotopy and the homology theory of weighted mappings
No full textWe compute the Hom functor of the homotopy category of weighted maps in terms of the cohomology groups of the domain with coefficients in the homology groups of the rang
The index bundle and bifurcation from infinity of solutions of nonlinear elliptic boundary value problems
No full textAbstract. We present three criteria for bifurcation from infinity of solutions of general boundary value problems for nonlinear elliptic systems of partial differential equations. Our sufficient conditions for bifurcation are computable, via the Atiyah-Singer family index theorem, from the coefficients of derivatives of leading order of the linearized differential operators and do not involve the analysis of the asymptotic derivative at infinit
Index bundle, Leray-Schauder reduction and bifurcation of solutions of nonlinear elliptic boundary value problems
No full textWe show that a family
of nonlinear elliptic
boundary value problems of index
parametrized by a compact manifold admits a reduction to a family of
compact vector
fields parametrized by if and only if its index bundle
vanishes. Our second conclusion is
that, in the presence of bounds for the solutions of the boundary value
problem,
the non vanishing of the image of the index bundle under generalized
-homomorphism produces
restrictions on the possible values of the degree of . The most
striking manifestation of
this arises when the first Stiefel-Whitney class of the index bundle is
nontrivial. In this case,
the degree of must vanish! From this we obtain a number of
corollaries about
bifurcation from infinity for solutions of nonlinear elliptic
boundary value problems
Orientation of Fredholm maps
No full textWe introduce a notion of orientation of a Fredholm map on a given compact subset of its domain and show that various approaches to orientation have as outcome the same class of orientable map
K-theoretic methods in bifurcation theory
No full textWe introduce to a new approach to bifurcation using the index bundle of the linearization at the trivial branch
- …
