Topological Transversality

Part of the Topological Fixed Point Theory and Its Applications book series (TFPT, volume 5)

We study bifurcation of chaotic solutions of discontinuous o.d.eqns in Chapter 4 using topological degree methods. Those results are extensions of a similar classical result, the Smale-Birkhoff homoclinic theorem for smooth o.d.eqns, based on the existence of Smale’s horseshoe which is a consequence of a transversal intersection of stable and unstable manifolds of a hyperbolic fixed point of a diffeomorphism. When the smoothness of an o.d.eqn is dropped, then this classical approach fails. For this reason we use topological degree arguments. This is the aim of Chapter 4. Similar mathematical difficulties occur when a diffeomorphism possesses a hyperbolic fixed point, but the corresponding stable and unstable manifolds do not have a transversal intersection. So a natural question arises that which kind of intersection should have stable and unstable manifolds in order to have a chaotic behavior of a diffeomorphism near that intersection. The aim of this section is to give an answer on this question by extending the Smale-Birkhoff homoclinic theorem in this direction. We show that a topologically transversal intersection of stable and unstable manifolds guaranties chaotic behavior of the diffeomorphism.


Periodic Orbit Periodic Point Unstable Manifold Homoclinic Orbit Exponential Dichotomy 
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© Springer Science + Business Media B.V. 2008

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