# A K-theoretical invariant and bifurcation for homoclinics of Hamiltonian systems

@article{Portaluri2016AKI, title={A K-theoretical invariant and bifurcation for homoclinics of Hamiltonian systems}, author={Alessandro Portaluri and Nils Waterstraat}, journal={Journal of Fixed Point Theory and Applications}, year={2016}, volume={19}, pages={833-851} }

We revisit a K-theoretical invariant that was invented by the first author some years ago for studying multiparameter bifurcation of branches of critical points of functionals. Our main aim is to apply this invariant to investigate bifurcation of homoclinic solutions of families of Hamiltonian systems which are parametrised by tori.

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## References

SHOWING 1-10 OF 28 REFERENCES

Bifurcation of Homoclinics of Hamiltonian Systems

- Mathematics
- 2008

We 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…

Spectral flow, crossing forms and homoclinics of Hamiltonian systems

- Mathematics
- 2014

We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of…

Bifurcation of homoclinics

- Mathematics
- 2008

We 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…

A family index theorem for periodic Hamiltonian systems and bifurcation

- Mathematics
- 2013

We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah–Singer index theorem for selfadjoint elliptic operators. For the special case of…

Spectral flow and bifurcation for a class of strongly indefinite elliptic systems

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2017

We consider bifurcation of solutions from a given trivial branch for a class of strongly indefinite elliptic systems via the spectral flow. Our main results establish bifurcation invariants that can…

A Morse-Smale index theorem for indefinite elliptic systems and bifurcation

- Mathematics
- 2014

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of…

A remark on bifurcation of Fredholm maps

- MathematicsAdvances in Nonlinear Analysis
- 2016

Abstract We modify an argument for multiparameter bifurcation of Fredholm maps by Fitzpatrick and Pejsachowicz to strengthen results on the topology of the bifurcation set. Furthermore, we discuss an…

Nonorientability of the index bundle and several parameter bifurcation

- Mathematics
- 1991

The universal parameter space for Sturm-Liouville boundary value problems is a two torus. We show that the index bundle of the universal family of linear S.L. boundary value problems is…

Bifurcation results for critical points of families of functionals

- Mathematics
- 2012

Recently the first author studied the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrised by a compact and orientable manifold having a non-vanishing…

A K-theoretical invariant and bifurcation for a parameterized family of functionals

- Mathematics
- 2009

Abstract Let F : = { f x : x ∈ X } be a family of functionals defined on a Hilbert manifold E ˜ and smoothly parameterized by a compact connected orientable n -dimensional manifold X , and let σ : X…