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

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.

💡 Research Summary

The paper investigates the bifurcation behavior of normally hyperbolic invariant manifolds (NHIMs) in a three-degree‑of‑freedom Hamiltonian system that models a test particle moving under the gravitational influence of two interacting dwarf galaxies. The authors adopt a rotating frame and construct an effective potential that combines the Newtonian attraction of the two spherical galaxies (with unequal masses m₁ = 0.06 and m₂ = 0.14) and the centrifugal term. The distance between the galaxy centers is set to R = 6 and the softening length to c = 0.25, keeping the total mass fixed at 0.20.

In the horizontal x‑y plane the effective potential exhibits three collinear Lagrange points (L₁, L₂, L₃), each an index‑1 saddle of the potential. These saddles give rise to three fundamental NHIMs of codimension two. Each NHIM possesses an internal dynamics that can be visualized on a two‑dimensional Poincaré map obtained by projecting the flow onto a canonical plane. The NHIMs are characterized by a normal (transverse) Lyapunov exponent and one or two tangential exponents; normal hyperbolicity holds as long as the normal exponent dominates.

The Jacobi constant E_J is used as the bifurcation parameter. By varying E_J the authors track how the NHIMs evolve, how their internal KAM tori break, and when the normal hyperbolicity is lost. For low E_J values the NHIMs are surrounded by robust families of invariant tori, and the projected Poincaré sections display regular islands. As E_J increases, several pitchfork bifurcations occur: each horizontal Lyapunov orbit emerging from a potential minimum undergoes a symmetry‑breaking pitchfork, producing a pair of asymmetric periodic orbits together with the original symmetric one. These bifurcations are marked by black squares in the bifurcation diagram.

At higher E_J the normal Lyapunov exponent of each NHIM approaches the tangential ones. When the two become comparable, the NHIM loses normal hyperbolicity (indicated by eight‑pointed stars). The KAM tori dissolve, and the NHIM fragments into lower‑dimensional invariant pieces. The “outer” remnants of the broken NHIM generate transient chaotic layers that act as conduits between the stable and unstable manifolds of the remaining NHIM fragments.

A striking result is the coordinated timing of these events across the three NHIMs. In the asymmetric mass configuration the loss of normal hyperbolicity of the middle NHIM (associated with L₂) occurs in essentially the same E_J interval as the pitchfork bifurcations of the outer NHIMs (associated with L₁ and L₃). This coordination is not a trivial consequence of symmetry; instead it arises from the intricate coupling of the manifolds in the full six‑dimensional phase space. The authors demonstrate this coupling using two diagnostic tools. First, they compute the projected Poincaré maps for each NHIM as E_J varies, revealing simultaneous creation, destruction, and stability changes of fixed points. Second, they evaluate a delay‑time function that measures how long an orbit launched near a manifold remains in its vicinity before escaping. Peaks in the delay‑time surface correspond to the stable manifolds of the NHIMs, while valleys trace the unstable manifolds. The delay‑time plots clearly show overlapping regions where the transient outer parts of one broken NHIM intersect the manifolds of the others, confirming that the transient dynamics mediate the coordination.

Numerical experiments with the chosen parameters produce the following characteristic energies: E_J₁ ≈ 0.06511, E_J₂ ≈ 0.05481, E_J₃ ≈ 0.05921 for the three collinear Lagrange points, and E_J₆ ≈ 0.2714, E_J₇ ≈ 0.5714 for the two potential minima near the galaxy centers. From each minimum three primary periodic orbits emerge: two horizontal (one with positive, one with negative rotation) and one mainly vertical. The negatively rotating horizontal orbits are the ones that undergo the pitchfork bifurcations at x ≈ 3, E_J ≈ 0.063 (from P₁) and x ≈ 2.8, E_J ≈ 0.074 (from P₂). The vertical orbits lose normal hyperbolicity around E_J ≈ 0.07–0.08, after which their outer fragments generate the transient chaotic layers described above.

The authors conclude that in a strongly asymmetric binary galaxy model the bifurcation scenarios of the three NHIMs are not independent; rather, they are mutually coordinated through the transient outer parts that appear when normal hyperbolicity is lost. This phenomenon had not been observed in earlier symmetric models and highlights the importance of transient chaos in high‑dimensional Hamiltonian dynamics. The findings have implications for transition‑state theory, reaction‑rate calculations, and the general understanding of transport in celestial mechanics. Future work is suggested to explore whether similar coordination occurs in other multi‑body systems and to develop rigorous computer‑assisted proofs of NHIM persistence and breakdown in such contexts.