Generating hyperbolic singularities in completely integrable systems

Abstract

Let $(M,\Omega )$ be a connected symplectic 4-manifold and let $F=(J,H):M\to {\mathbb{R}}^2$ be a completely integrable system on $M$ with only non-degenerate singularities and for which $J:M\to \mathbb{R}$ is a proper map. Assume that $F$ does not have singularities with hyperbolic blocks and that $p_1,...,p_n$ are the focus-focus singularities of $F$. For each subset $S=\{i_1,...,i_j\}$ we will show how to modify $F$ locally around any $p_i,i\in S$, in order to create a new integrable system $\tilde{F}=(J,\tilde{H}):M\to {\mathbb{R}}^2$ such that its classical spectrum $\tilde{F}(M)$ contains $j$ smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of $\tilde{F}$. Moreover the focus-focus singularities of $\tilde{F}$ are precisely $p_i$, $i\in \{1,...,n\}\setminus S$, and each of these $p_i$ is non-degenerate. The proof is based on Eliasson's linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.

24 pages, 9 figures

References in corpus (3)

• Vanishing Twist in the Hamiltonian Hopf Bifurcation
• Symplectic bifurcation theory for integrable systems
• (Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation

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