Hodge Coercivity and Global Dynamics in Two-Field Edge-Cochain Systems with MHD-Type Cancellation

The graph-Hodge literature supplies several relevant, but distinct, points of comparison. A Hodge decomposition separates edge fields into gradient, curl, and harmonic parts; Lim [11] emphasizes that this structure can be developed by linear algebra on incidence matrices, while Horak and Jost [9] provide a spectral framework for combinatorial Laplacians on simplicial complexes. The present phase space starts after the gradient component has been removed: $\mathcal{V}=\ker d_{0}^{*}$ is the divergence-free edge space, and the remaining splitting is between a dissipative complement and harmonic circulation directions.

Many existing dynamical uses of the Hodge Laplacian are linear. The normalized $1$-Laplacian of Schaub et al. [14] governs edge-space random walks, and the balanced Hodge Laplacians of Ziegler, Hlinka, and Thurner [17] quantify how lower- and higher-order interactions affect edge consensus. Higher-order network work more generally uses simplicial complexes to represent interactions that cannot be reduced to pairs [4, 3]; nonlinear synchronization on simplicial complexes provides another distinct line of development [12]. The present problem differs in two respects. It is a nonlinear two-field dissipative evolution rather than a diffusion, consensus, or phase-synchronization equation, and the key question is not simply spectral convergence but the availability of a global energy estimate after the nonlinear exchange terms have cancelled. Thus the role of the Hodge Laplacian here is not only spectral smoothing or consensus formation, but the creation, or failure, of a coercive energy mechanism after nonlinear cancellation.

This distinction clarifies the novelty boundary. The paper does not introduce a new general theory of Hodge Laplacians, nor does it claim a physical transport discretization. It identifies a class of nonlinear constrained cochain systems for which the Hodge kernel has a direct dynamical consequence: it is exactly the obstruction to the standard full-phase-space absorbing-set proof. The harmonic-fibre theorem below then shows how a further algebraic condition can replace full-space coercivity by fibre-wise coercivity.

The decomposition in (2) is orthogonal and preserved by the linear Hodge semigroup. If $\{\phi_{j}\}$ is an orthonormal eigenbasis of $\Delta_{1}|_{\mathcal{V}}$ with eigenvalues $\lambda_{j}\geq 0$, then the harmonic vectors are precisely the modes with $\lambda_{j}=0$, and the remaining modes satisfy $\lambda_{j}\geq\lambda_{+}$. In the linear, unforced limit, the components of a solution satisfy

Consequently, the Hodge Laplacian alone leaves $Qu$ and $Qh$ unchanged and exponentially damps the complementary components. This simple modal calibration is useful because it separates two issues that are sometimes conflated: a nontrivial harmonic space is a spectral obstruction to full-space coercivity, whereas its actual nonlinear evolution depends on the interaction map.

The restricted positive eigenvalue has a direct variational characterization,

Thus $\lambda_{+}$ is not merely a numerical quantity extracted from a matrix; it is the exact constant that controls every complementary-space energy estimate. The disk, annulus, and two-hole examples below have harmonic dimensions zero, one, and two, respectively. Their spectra therefore provide a direct computational realization of the alternative in Theorem 1.

The energy argument uses only the range condition and (4); it does not use a coordinate representation of $B$. This separation is important. Let

Then the cancellation identity is equivalently

The nonlinear vector field is therefore energy-neutral, even though the individual field equations exchange energy through both self- and cross-couplings. This is the finite-dimensional analogue of the total-energy cancellation that motivates the MHD-type terminology.

The cancellation class is stable under two useful operations. If $B_{1}$ and $B_{2}$ satisfy (4), then every linear combination $\alpha B_{1}+\gamma B_{2}$ does as well. Further, let $R$ be an orthogonal projector on $\mathcal{V}$ and let $\widetilde{B}$ be admissible. On the restricted phase space $R\mathcal{V}$, define $B_{R}(a,b):=R\widetilde{B}(a,b)$ for $a,b\in R\mathcal{V}$. For $c\in R\mathcal{V}$, orthogonality gives $\langle B_{R}(a,b),c\rangle=\langle\widetilde{B}(a,b),c\rangle=-\langle\widetilde{B}(a,c),b\rangle=-\langle B_{R}(a,c),b\rangle$. Thus $B_{R}$ satisfies the same trilinear antisymmetry on $R\mathcal{V}$. The harmonic-decoupled construction in Section 5 is a further projected construction that also removes harmonic dependence. These elementary closure properties show that the cancellation assumption specifies an interaction class rather than a single engineered matrix formula.

For the anticommutator realization, bilinearity also gives an explicit local Lipschitz estimate. If $a,b,c,d\in\mathcal{V}$, then

and therefore

The estimate makes the continuous-semiflow formulation below quantitative on every bounded set. It is not, by itself, a derivative-of-semiflow estimate sufficient for a nontrivial attractor-dimension bound.

For the realization in Proposition 1, the right-hand side of (5)–(6) is polynomial and locally Lipschitz. The cancellation identity prevents finite-time blow-up even when the complex is not coercive.

The solution map $S(t)$ is therefore a continuous semiflow on $\mathcal{V}\times\mathcal{V}$. This elementary observation is used below only to apply the standard absorbing-set construction; no nontrivial attractor-dimension estimate is claimed.

For the explicit realization, continuous dependence can be quantified. Let $(u,h)$ and $(\bar{u},\bar{h})$ be two solutions with the same forcing, and assume that all four component norms remain bounded by $R$ on $[0,T]$. Bilinearity and (8) imply

and the same estimate holds for each mixed difference after replacing one factor by its difference. Consequently, if

there is a constant $C_{J}>0$, depending only on $J$, such that

A direct Euclidean-norm estimate may take $C_{J}=16\|J\|$. The dissipative terms have simply been dropped in this upper bound. Grönwall’s inequality yields

Although (11) is not a contraction estimate, it makes explicit why the polynomial system defines a continuous semiflow and why the attractor construction can be based on a compact absorbing ball. Obtaining a useful volume-contraction estimate would require sharper information about the derivative of the nonlinear flow.

The scalar estimate supplies more than boundedness. Integrating the exact identity gives, for every $T>0$,

Thus a bounded forced trajectory has bounded time-averaged dissipation. More specifically, Proposition 2 and Cauchy–Schwarz show that the mean dissipation is controlled entirely by the projected forcing magnitude and the absorbing radius. This observation is elementary in finite dimensions, but it explains why the numerical energy diagnostic is meaningful: the pointwise cancellation law is exact, while numerical quadrature only approximates the integral in (16).

The forcing is deliberately time independent here because it generates an autonomous semiflow. Time-dependent forcing would lead to a nonautonomous process and requires a pullback- or uniform-attractor framework, which is outside the current scope. Likewise, a quantitative fractal-dimension estimate would require control of the linearized flow on the absorbing set. Neither follows from energy cancellation alone.

The theorem isolates a genuine mechanism rather than concealing the noncoercive difficulty. In the general system, the right-hand sides of (17)–(18) contain quadratic terms that can change both the harmonic coordinates and the complementary energy. Thus the unrestricted system may exhibit harmonic exchange, as demonstrated below on a two-hole complex. The theorem does not claim that all admissible interactions have bounded full-space attractors in this case.

By contrast, $B_{0}$ removes every nonlinear input to the harmonic equation and every dependence of the complementary equation on the fixed harmonic coordinates. The restricted system is therefore not merely a projected simulation: it is a well-defined model on each affine fibre. In the unforced case its attractor is the single harmonic state $(u_{H},h_{H})$ in that fibre. In the forced case, the complementary forcing produces a compact fibre-wise attractor, while the prescribed harmonic state remains unchanged. This is the precise sense in which harmonic directions can be controlled without introducing an artificial damping term.

A different extension would add harmonic damping terms $-\gamma Qu$ and $-\gamma Qh$ with $\gamma>0$. Such a model restores a coercive full-space linear operator and is suitable when decay of harmonic circulation is intended. It is not adopted here because it changes the dynamics rather than characterizing the topology-induced neutral directions of the original Hodge-Laplacian system.

The fibre construction can be written in coordinates that make its spectral content transparent. Fix $(u_{H},h_{H})\in\mathcal{H}^{1}\times\mathcal{H}^{1}$ and write

For the interaction $B_{0}$, the equations for $(u_{0},h_{0})$ are

No coefficient in the two reduced equations (21) depends on $(u_{H},h_{H})$. Hence all unforced harmonic fibres carry isomorphic complementary dynamics, translated by their fixed harmonic labels. In particular, the linearization at the unforced fibre equilibrium $(u_{H},h_{H})$ is

Its spectrum lies in $(-\infty,-\beta_{+}]$, with $\beta_{+}=\min(\nu,\eta)\lambda_{+}$. Although the nonlinear terms modify the modal dynamics, their exact energy cancellation allows the linear coercive decay estimate to extend globally to the nonlinear complementary system.

This reduction also explains the numerical comparison in Figure 4. The left-hand panel is generated by the original admissible realization, for which the projected equations contain harmonic coupling terms. The right-hand panel is generated by (21). The two runs therefore compare two mathematically distinct, explicitly specified systems rather than two parameter choices within one unexamined simulation.

Three complexes are used. The triangulated disk is a six-sector fan with trivial first cohomology. The triangulated annulus has one harmonic direction. The third complex is a deterministic $6\times 4$ triangulated rectangle with two separated unfilled squares. It has two harmonic directions and a nontrivial complementary space $\mathcal{V}_{0}$. Unlike an unfilled single cycle, whose divergence-free space can be entirely one-dimensional and harmonic, this two-hole example has enough room to display a meaningful distinction between harmonic and dissipative degrees of freedom. Figure 1 gives the geometries, and Figure 2 displays the restricted Hodge spectra. The red zero modes in the latter are exact topological kernel directions up to the stated numerical tolerance.

For an edge space of dimension $m$, the fixed skew-symmetric matrix has nearest-neighbour entries

with closing entries $J_{0,m-1}=-0.17$ and $J_{m-1,0}=0.17$. The baseline parameters are $\nu=0.55$ and $\eta=0.35$. Projected trigonometric arrays provide deterministic initial states.

Before integration, the script checks the symmetry and idempotence of $P$, the relation $d_{0}^{*}P=0$, skew-symmetry of $M(a)$, trilinear antisymmetry, and total nonlinear energy cancellation. For the two-hole runs, the largest algebraic residual is below $1.34\times 10^{-15}$. On the disk, Figure 3 compares unforced exponential decay with a bounded forced response. The right panel reports

The integral is evaluated by composite Simpson quadrature on 16,001 output times. The maximum normalized residual is $3.82\times 10^{-10}$, making explicit that this is an a posteriori quadrature-and-solver diagnostic rather than the exact analytical cancellation statement of Theorem 2.

Figure 4 addresses the noncoercive issue directly. In the left panel, the corrected interaction from Proposition 1 permits a nonzero change in harmonic energy on the two-hole complex; for the displayed deterministic trajectory its maximum change is $2.57\times 10^{-3}$. This illustrates the transfer mechanism in (17)–(18) without implying a universal residual-energy law. In the right panel, the harmonic-decoupled interaction from (19) keeps the harmonic energy constant to a maximum change of $5.14\times 10^{-16}$ while the complementary energy decays. The companion algebraic residual for harmonic decoupling is below $4.27\times 10^{-16}$. Thus the computation distinguishes the general noncoercive case from the structured fibre-wise result of Theorem 5.