Normally Hyperbolic Invariant Manifolds (NHIMs)

• NHIMs are invariant manifolds with dominant normal expansion or contraction, serving as organizing structures in high-dimensional Hamiltonian dynamics.
• They remain robust under small perturbations but decay when tangential instabilities overcome normal contraction, leading to fragmented and chaotic regions.
• Time delay functions and Poincaré maps effectively visualize NHIM decay and quantify transient trapping in complex dynamical systems.

A Normally Hyperbolic Invariant Manifold (NHIM) is an invariant subset of phase space whose normal (transverse) dynamics expand or contract at rates that dominate any instabilities or contraction tangent to the manifold. In three-degree-of-freedom (3-dof) Hamiltonian systems, such as the perturbed magnetic dipole model for electron motion, NHIMs of codimension-2 act as central organizing structures for transport and transient behavior. The decay of these structures under perturbation, the interplay of tangential and normal instabilities, and their detection via indicator functions and Poincaré maps form the focus of contemporary research in high-dimensional Hamiltonian dynamics.

1. Structural Role of NHIMs in 3-DoF Hamiltonian Systems

In a 3-dof Hamiltonian system, the phase space is six-dimensional, and restriction to a constant energy shell yields five-dimensional dynamics. The Poincaré map, constructed on an appropriate four-dimensional section, reveals invariant subsets (NHIMs) that typically appear as two-dimensional invariant surfaces in the domain of the map. These NHIMs are robust under small perturbations and serve as skeletons for transport across index-1 saddles, guiding orbits between dynamically distinct regions.

For example, in the unperturbed setting (such as an electron in an axisymmetric magnetic dipole field), a continuous family of Lyapunov orbits—parameterized by conserved angular momentum—pile up into a NHIM cylinder. This surface persists under perturbation as long as normal expansion/contraction rates exceed all tangential Lyapunov exponents.

2. Decay Scenario: Loss of Normal Hyperbolicity

As the system is perturbed (for instance, by introducing a magnetic quadrupole term), the NHIM begins to decay through a loss of normal hyperbolicity:

• Persistence Threshold: The classical Fenichel theorem ensures NHIM persistence if, everywhere on the manifold, the normal Lyapunov exponents $\lambda_n$ strictly exceed those tangent to the manifold $\lambda_t$.
• Onset of Decay: If, locally, $\lambda_t > \lambda_n$, normal hyperbolicity fails and the invariant manifold ceases to exist as a smooth entity at those points.
• Partial Survival: The breakdown is rarely instantaneous; surviving "islands" of the NHIM (often corresponding to KAM tori) coexist with regions of fractalized remnants and chaotic layers. In the extremal limit, the NHIM becomes a "fractal powder," with only transient influence on passing trajectories.
• Spatial Manifestation: In the Poincaré map, this manifests as the breakup of a smooth, continuous surface into disjoint clusters or bands. Surviving KAM curves organize the remaining structure, separated by thin longitudinal chaos strips where tangential instability dominates.

3. Tangential Transient Effects and Their Detection

When the NHIM decays, a new phenomenon emerges—tangential transients:

• Pre-decay: In the unbroken NHIM, orbits near the manifold depart solely in the normal direction, quickly escaping the neighborhood.
• Post-decay: Once decay has begun, orbits can leave tangentially along the boundary of the NHIM—first lingering near the surviving portion and then sliding off along the tangent before fully escaping. These tangential transients produce a characteristically smooth, high (but finite) background in various phase space indicators.
• Physical implication: This mechanism implies that, even after the mathematical breakdown of the NHIM, the system retains regions with enhanced transient trapping and sticky dynamics, influencing overall transport rates.

4. Internal Poincaré Maps: Visualization and Quantification

The restricted (internal) Poincaré map effectively visualizes the NHIM's internal dynamics and its breakup:

• Construction: The map is built by intersecting the NHIM with a four-dimensional section (e.g., $z=0$), then projecting onto relevant coordinates (such as $(\varphi, L_z)$ for the electron model).
• Analysis in Unperturbed Case: For $\epsilon=0$, the map is foliated by invariant horizontal lines—a reflection of the conserved angular momentum. Each line corresponds to a regular, non-chaotic motion.
• Under Perturbation: As symmetry breaks (nonzero $\epsilon$), invariant lines split into island chains and chaotic layers. Surviving KAM curves indicate intact regions of the NHIM, while chaotic separatrices identify breakup zones and tangential decay fronts.
• Decay Front: The boundary beyond which the NHIM fragments corresponds to the locus where normal and tangential Lyapunov exponents cross.

5. Time Delay Indicator Functions: Quantitative Detection of Decay

A time delay function $t_d$—the sum of forward and backward delay times—serves as an efficient and sensitive indicator for NHIM decay and the presence of tangential transients:

$$t_d = t_d^+ + t_d^- = \tau - \frac{r^+ - r_0}{v^+} + \tau - \frac{r^- - r_0}{v^-}$$

where $r_0$ is the initial position, $r^\pm$ are positions after time $\pm\tau$, and $v^\pm$ the corresponding asymptotic velocities.

• Intersections with Stable Manifolds: At points where initial conditions lie exactly in the stable manifold of the (surviving) NHIM, $t_d$ diverges, signaling a singularity in the indicator map.
• Tangential Transients: Regions affected by the decay of the NHIM show high but finite $t_d$—providing a "halo" of delayed escape associated with orbits that temporarily follow the decaying NHIM before tangential ejection.
• Fractal Structure: As decay proceeds, the set of initial conditions with large $t_d$ (i.e., long escape times) exhibits a fractal structure, which encodes the skeleton of surviving and remnant invariant sets.
• Practical computation: Delay time plots on planes such as $(r, L_z)$ at fixed $\varphi$ and $p_r$ directly encode the global location and degree of decay of the NHIM and can differentiate true intersections (singularities) from tangentially transient regions (diffuse high background).

6. Case Study: Electron in a Perturbed Magnetic Dipole Field

The decay scenario is exemplified by the electron in a perturbed dipole field:

• Hamiltonian: In cylindrical coordinates, with the perturbation parameter $\epsilon$,

$$H = \frac{1}{2m} \left( p_r^2 + \left( p_\varphi - A_0 \frac{r}{(r^2+z^2)^{3/2}} \right)^2 + \left( p_z - \epsilon \frac{r^2 \sin{2\varphi}}{(r^2 + z^2)^{5/2}} \right)^2 \right)$$

• Unperturbed ($\epsilon = 0$): Rotational symmetry yields conservation of $L_z$, and the NHIM is a continuous cylinder over the family of reduced Lyapunov orbits.
• Perturbed ($\epsilon \neq 0$): The continuous NHIM breaks up as the quadrupole perturbation destroys symmetry. Decay begins immediately at the NHIM boundary, corresponding to the locus where normal instability vanishes.
• Indicator Function View: Time delay plots exhibit clear singularities (surviving stable manifolds) and high diffuse regions (tangential decay). The transition is also evident in restricted Poincaré maps, where KAM islands fragment and chaos layers expand.

7. Synthesis: Combined Geometric and Quantitative Detection of NHIM Decay

By simultaneously analyzing the NHIM's restricted Poincaré map (capturing the fate of invariant curves, chaotic zones, and island chains) and the time delay function (encoding global trapping, singularities, and the onset of tangential transients), a comprehensive picture of the decay scenario emerges:

• Operator-Level Picture: Decay is controlled by the interplay of Lyapunov exponents in normal vs. tangential directions, manifested both geometrically (Poincaré map breakup) and quantitatively (indicator function singularities/broad backgrounds).
• Practical Outcome: Time delay (or related phase space indicator) functions are computationally efficient tools for revealing the onset, nature, and spatial extent of NHIM decay and transient trapping, even when direct visualization of high-dimensional invariant sets is infeasible.
• Broader Implications: These approaches generalize to other high-dimensional Hamiltonian systems, offering mechanisms to detect, quantify, and interpret the breakdown of invariant geometric structures crucial for long-time transport and global phase space organization.

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This synthesis illustrates the decay of codimension-2 NHIMs under perturbation, the manifestation of tangentially transient behavior, and the power of combined geometric and indicator-function approaches for exploring the structure and fate of invariant manifolds in complex dynamical systems (Montoya et al., 3 Jan 2025).