Why fourth-order invariants are not unique
The square-matrix method builds an approximate invariant order by order from the one-turn map of a non-integrable Hamiltonian system. The coefficient vector of the invariant is the eigenvector of the transposed square matrix associated with the unit eigenvalue, and each higher order is obtained by solving a homological equation of the form (I - m3) V = m2 V_lower.
At fourth and higher even orders this equation becomes singular. The rotationally invariant polynomial (x^2 + p^2)^2 = x^4 + 2 x^2 p^2 + p^4, with coefficient vector n = [1, 0, 2, 0, 1]^T, spans the null space of I - m3. The fourth-order invariant is therefore determined only up to an arbitrary multiple of (x^2 + p^2)^2, written W(4) = W(4,0) + c4 (x^2 + p^2)^2, and c4 cannot be fixed from the fourth-order homological equation alone. The paper notes that if the quadratic invariant carries a nonzero cross term (alpha not equal to 0), the null vector must be modified to [gamma^2, 4 alpha gamma, 2 beta gamma + 4 alpha^2, 4 alpha beta, beta^2].
Minimum-norm solution reproduces the central KAM region
One way to pick a representative is the minimum-norm solution. The paper removes one linearly dependent row from the homological equation and adds the orthogonality constraint n^T V(4) = 0, forcing the solution to be orthogonal to the null space. This yields a unique representative, W(4,0), but the paper stresses that it provides no systematic physical criterion for choosing the null-space contribution appropriate to resonant dynamics.
The comparison in the paper makes the consequence concrete. Setting the null-space coefficients c*_{4,6,8,...} = 0 selects the minimum-norm representatives without null-space corrections. That choice still reproduces the central KAM region, but it fails to capture the resonance-island topology.
Averaging over a region of interest fixes the null space
The method proposed here keeps the recursive square-matrix structure but determines the null-space coefficients by an averaging procedure. The optimal coefficients are selected by minimizing the spatial variation of the leading-order residual of the approximate invariant over a prescribed region of interest (RoI). The null-space basis vectors are incorporated explicitly, and their coefficients are chosen so that the resulting invariant reproduces the resonance-island topology.
The demonstration uses the Kobayashi map, a linear symplectic rotation followed by a thin nonlinear kick, with parameters beta = gamma = 1, alpha = 0, mu = 2 pi (1/3 + 1/100), and b4 = 1/10. This places the tune near a third-order resonance, where three stable resonance islands are present. Even when the tune crosses the third-order resonance, the motion remains bounded over the simulated interval and cycles among the three stable islands.
Null-space corrections restore the three resonance islands
The decisive comparison is between eighth-order invariants with and without the null-space corrections, checked against tracking simulations. Without the corrections, the invariant contours reproduce the central KAM region but miss the resonance-island topology. With the null-space coefficients determined by the averaging procedure, the invariant contours follow the tracking results and recover the island structure.
This also explains why the ambiguity was not apparent in the earlier application to the NSLS-II storage ring. Its tune is sufficiently far from low-order resonances, and the central KAM region is already well reproduced without explicitly accounting for the null-space contributions. The null-space ambiguity is algebraically generic and present irrespective of resonance; its physical consequence becomes important only near a low-order resonance, where different representatives of a truncated approximate invariant can describe the resonance-island topology very differently.
Koopman and square-matrix invariants differ only by null space
The paper compares its sixth-order coefficients with those of a Koopman-based pseudo-Hamiltonian, normalized to the same quadratic Courant-Snyder invariant. The two constructions agree exactly through cubic order: x^3 = -0.277877, x^2 p = -0.100000, and x p^2 = 0.780012 in both. Higher-order coefficients are close but not identical; for example, x^4 is -2.868827 for Koopman and -2.822153 for the square-matrix construction, and p^4 is -3.226939 versus -3.180264.
The fourth-order difference follows the 1 : 2 : 1 ratio associated with (x^2 + p^2)^2: Delta C_x4 : Delta C_x2p2 : Delta C_p4 = 0.046674 : 0.093348 : 0.046674. The paper interprets this as evidence that the remaining difference lies within the intrinsic null-space freedom of the recursive construction rather than reflecting a change in the underlying resonance-island topology, and notes that the corresponding invariant contours differ only weakly over the RoI.
Invariant contours are not Hamiltonian flow
The paper draws a distinction that matters for how these objects can be used. An invariant of a discrete symplectic map constrains the map to approximately preserve its level sets, but it does not uniquely determine how points move along or between them. Treating the invariant as a continuous Hamiltonian and integrating Hamilton's equations produces trajectories that remain confined to individual resonance islands, whereas the one-turn map cyclically permutes the three islands and the M^3 map maps each island onto itself.
The invariant contours reproduce the resonance-island topology, but the invariant does not contain the discrete island-to-island permutation when interpreted as Hamiltonian flow. This limits direct application of the invariant-flow picture near this stable third-order resonance.
Where the method transfers and where it does not
The numerical demonstrations are intentionally restricted to one degree of freedom, where the null-space ambiguity and its resolution through averaging can be isolated without the complications of coupled resonances and high-dimensional phase-space topology. The algebraic construction extends formally to multiple degrees of freedom, but the paper leaves the coupled resonant case to future work. For two-degree-of-freedom transverse motion, the quartic-order null space is three-dimensional, spanned by W_x^2, W_x W_y, and W_y^2, with W_u = gamma_u u^2 + 2 alpha_u u p_u + beta_u p_u^2, and the averaging criterion would become a joint optimization over c_xx W_x^2 + c_xy W_x W_y + c_yy W_y^2.
Two special cases require extensions. If the linear tune lies exactly on a resonance, the null space of I - m3 can acquire additional dimensions and require additional null-space basis vectors. For the integrable McMillan-like map with a saddle fixed point at the origin, the Courant-Snyder invariant is not well defined at small amplitudes. The paper places both beyond the scope of conventional accelerator lattice design and optimization.
When the null-space choice changes the physics
The practical dividing line in this paper is not the presence of the even-order singularity, which is generic, but whether the detailed resonant topology is the object of interest. For most circular accelerators operated away from low-order resonances, the ambiguity has limited practical impact on the central KAM region, which is typically the phase-space structure of primary interest, and machine tunes are deliberately chosen to avoid low-order resonances. Stable resonance islands are instead exploited in specialized applications such as resonant extraction and resonance-island-based synchrotron-radiation techniques, which are often dominated by a single-plane resonance where a one-degree-of-freedom reduction provides the leading description.
For those island-dependent applications, the choice of null-space contribution is the decision that matters: the minimum-norm representative reproduces the central KAM region but not the islands, while the averaging procedure over a prescribed region of interest recovers the three stable islands of the Kobayashi map near third-order resonance. The paper also notes that approximate invariants are not unique even within a given construction, since alternative residual measures or different regions of interest could lead to different invariants.
Sources used in this answer
Construction of Approximate Invariants Across Nonlinear Resonances
Explicitly incorporating null-space basis vectors and fixing their coefficients by averaging over a prescribed region of interest resolves the non-uniqueness of recursively constructed approximate invariants and reproduces the three stable resonance islands of the Kobayashi map near a third-order resonance.