The NHIM bifurcation scenario of a particle in an asymmetric binary system of dwarf galaxies

TL;DR

Addresses a $3$-dof Hamiltonian model of a test particle in the asymmetric binary-dwarf galaxy potential and analyzes the bifurcation of three codimension-2 NHIMs associated with the index-1 saddles as the Jacobi constant $E_J$ varies. The study uses projected Poincaré maps, the delay-time indicator, and a stabilization-based algorithm to map the NHIMs and their internal dynamics, revealing coordinated bifurcations and the onset of transient chaos. The main findings include pitchfork and inverse-pitchfork sequences that break NHIMs in a way that generates chaotic seas and transient transport, with outer NHIMs showing coordinated evolution while the middle one displays distinct behavior. The results offer insights into phase-space transport in galactic dynamics and provide diagnostic tools applicable to high-dimensional Hamiltonian systems.

Abstract

We study the bifurcation scenario of a three-degree-of-freedom Hamiltonian system, a model based on the Lagrange restricted 3-body problem: a test particle moving in the gravitational field of a pair of interacting dwarf galaxies. The phase space of this system has 3 fundamental normally hyperbolic invariant manifolds (NHIMs) and their invariant stable and unstable manifolds form homoclinic/heteroclinic tangles. As the perturbation parameter increases, the NHIMs begin to lose normal hyperbolicity and their constituent KAM tori break, creating transient chaotic dynamics around them. We also observe a certain kind of coordination between the bifurcation scenarios of these NHIMs. We analyse this phenomenon using Poincaré maps and the delay time function.

Figures (14)

• Figure 1: The effective potential $V_{eff}$ in the horizontal plane. The green, blue and red curves are the equipotential lines to the energies $E_{J1}$, $E_{J2}$, and $E_{J3}$, respectively. Some additional equipotential curves are drawn as grey lines. The black dots mark the extremal points of the potential.
• Figure 2: Bifurcation diagram of the most important periodic orbits. On the panel (a), red curves represent horizontal orbits, dark green curves represent vertical orbits, blue curves represent tangentially stable tilted loop orbits, and the magenta curve represents tangentially unstable tilted loop orbits. Pitchfork bifurcations are marked as open black squares, and the loss of normal hyperbolicity is marked by eight-pointed black stars. On the panel (b), black curves represent periodic orbits which are unstable in the normal direction and stable in the tangential direction. Red curves represent periodic orbits which are unstable in the normal direction and also unstable in the tangential direction, where, however, the normal instability is larger than the tangential instability. Violet curves represent periodic orbits with larger tangential instability. And green curves represent periodic orbits outside of the NHIMs.
• Figure 3: Projected Poincaré map of the NHIM $\mathcal{M}^1_{E_J}$ and its remnant with transient trajectories for different values of the Jacobi constant $E_J$
• Figure 4: Projected Poincaré map of the NHIM $\mathcal{M}^3_{E_J}$ and its remnant with transient trajectories for different values of the Jacobi constant $E_J$
• Figure 5: Projected Poincaré map of the NHIM $\mathcal{M}^2_{E_J}$ for Jacobi constant $E_J = -0.035$.