Abstract
In the present paper, we describe a scenario of a disappearance of a non-compact heteroclinic curve for a three-dimensional diffeomorphism. As a consequence, it is established that 3-diffeomorphisms with a unique heteroclinic curve and fixed points of pairwise different Morse indices exist only on the 3-sphere. The described scenario is directly related to the reconnection processes in the solar corona, the mathematical essence of which, from the point of view of the magnetic charging topology, consists of a disappearance or a birth of non-compact heteroclinic curves.
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Notes
A support \(supp\{f_t\}\) of an isotopy \(\{f_t\}\) is the closure of the set \(\{x\in X: f_t(x)\ne f_0(x)\,\hbox {for some}\,t\in [0,1]\}\).
In Franks’ lemma, in a neighborhood \(U_p\) of the fixed point p of the diffeomorphism \(f:M^n\rightarrow M^n\) we consider the local chart \((U_p,\psi _p)\) where \(\psi ^{-1}_p=exp:T_xM^n\rightarrow U_p\) – exponential map. Then in these local coordinates the diffeomorphism f has the form \({\hat{f}}=exp^{-1}\circ f\circ exp:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\). The Franks lemma is that in any neighborhood of a diffeomorphism f there exists a diffeomorphism g having a fixed point p and a linear local representation \({\hat{g}}=exp^{-1}\circ g\circ exp\) if it is close enough to \(Df_p\). Thus, in any neighborhood of the diffeomorphism f there exists a diffeomorphism g, having a fixed point p and a linear local representation given by a matrix all of whose eigenvalues are pairwise different.
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This work was supported by the Russian Science Foundation (project 22-11-00027), except section 4, which was suppported by the Laboratory of Dynamical Systems and Applications (project 075-15-2022-1101).
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Pochinka, O.V., Shmukler, V.I. & Talanova, E.A. Bifurcation of a disappearance of a non-compact heteroclinic curve. Sel. Math. New Ser. 29, 60 (2023). https://doi.org/10.1007/s00029-023-00863-w
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DOI: https://doi.org/10.1007/s00029-023-00863-w