Double flip bifurcations in $\mathbb{Z}/2\mathbb{Z}$-symmetric Hamiltonian systems

TL;DR

The paper introduces the double flip bifurcation in $bZ_{2}$-symmetric Hamiltonian systems, occurring when a two-parameter family $H_{j,t}$ is reduced to 1-DOF and a bifurcation in $t$ induces two Hamiltonian flip bifurcations in $j$. It establishes a universal 6-jet normal form $H_{j,t}(q,p)=rac{a}{2}p^{2}+rac{b}{6}q^{6}+rac{ u_{1}(j,t)}{2}q^{2}+rac{ u_{2}(j,t)}{4}q^{4}$ under precise nondegeneracy assumptions, and shows that the two $j$-bifurcations occur at the zeros of $ u_{1}$, with their duality determined by $a u_{2}$. The bifurcation diagram is analyzed via discriminants $ ext{disc}_{b}(j,t)$ and stability of critical points, yielding a criterion for when the two saddle-node bifurcations are concave in the same or opposite directions, and hence whether the double flip is degenerate. The paper validates the theory through three concrete examples—the $(1:-2)$ oscillator, a modified oscillator, and the Hirzebruch surface $W_{2}(1,1)$—each illustrating the predicted two-branch Hamiltonian flip structure and the non-degenerate opposite-concavity geometry of the bifurcations. Overall, the work provides a normal-form framework and diagnostic tools for a new class of Hamiltonian bifurcations with $bZ_{2}$ symmetry and two-parameter unfolding.

Abstract

In this paper we introduce a new bifurcation in Hamiltonian systems, which we call the double flip bifurcation. The Hamiltonian depends on two parameters, one of which controls the double flip bifurcation. The result of the bifurcation is the occurrence of two Hamiltonian flip bifurcations with respect to the other parameter. The two Hamiltonian flip bifurcations are simultaneous with respect to the first parameter, and are connected by a curve-segment of singular points. We find a normal form for Hamiltonians describing systems going through double flip bifurcations, and compute said normal form for some examples.

Figures (10)

• Figure 1:
• Figure 2:
• Figure 3: The grey area is the cone $\mathcal{B}$ projected to the plane $\{w=0\}$, and the blue curves are level sets $H_{j,t}^{-1}(h)$ for various levels $h$. The level set $H_{j,t}^{-1}(0)$ is drawn as a dashed curve. The top row shows a Hamiltonian flip bifurcation, as in the middle-figure $H_{j,t}^{-1}(0)$ only intersects $\mathcal{B}$ at its apex. The bottom row shows a dual Hamiltonian flip bifurcation, as in the middle-figure $H_{j,t}^{-1}(0)$ has further intersections with $\mathcal{B}$.
• Figure 4: Here $a = b = 1$, and $\nu_{1}(j,t) = j^{2}-t$, $\nu_{2}(j,t) = 3j-t$.
• Figure 8: Projections of the reduced phase space onto the $Y=0$ plane. To the left it has a conical singularity at the bottom, in the middle a cuspidal singularity, and to the right no singularity.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Definition 3.1
• Definition 3.2
• Theorem 3.3: Gibson1979
• Lemma 3.4