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Pré-Publication, Document De Travail Année : 2016

Homoclinic bifurcation in Morse-Novikov theory a doubling phenomenon

Résumé

We consider a compact manifold of dimension greater than 2 and a differential form of degree one which is closed but non-exact. This form, viewed as a multi-valued function has a gradient vector field with respect to any Riemannian metric. After S. Novikov's work and a complement by J.-C. Sikorav, under some genericity assumptions these data yield a complex, called today the Morse-Novikov complex. Due to the non-exactness of the form, its gradient has a non-trivial dynamics in contrary to gradients of functions. In particular, it is possible that the gradient has a homoclinic orbit. The one-form being fixed, we investigate the codimension-one stratum in the space of gradients which is formed by gradients having one simple homoclinic orbit. Such a stratum S breaks up into a left and a right part separated by a substratum. The algebraic effect on the Morse-Novikov complex of crossing S depends on the part, left or right, which is crossed. The sudden creation of infinitely many new heteroclinic orbits may happen. Moreover, some gradients with a simple homoclinic orbit are approached by gradients with a simple homoclinic orbit of double energy. These two phenomena are linked.
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Dates et versions

hal-01304121 , version 1 (19-04-2016)
hal-01304121 , version 2 (05-07-2016)
hal-01304121 , version 3 (28-05-2018)
hal-01304121 , version 4 (20-05-2019)

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François Laudenbach, Carlos Moraga Ferrándiz. Homoclinic bifurcation in Morse-Novikov theory a doubling phenomenon. 2016. ⟨hal-01304121v4⟩
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