Nonlinearity Index (NI) Across Domains

• Nonlinearity Index (NI) is a domain-specific diagnostic that quantifies nonlinear behavior using measures like harmonic ratios, sensitivity metrics, or power-law exponents.
• In solar physics, NI is computed from the ratio of harmonic power between fundamental and second harmonics, indicating wave steepening and shock formation.
• Other applications include solution-dependent indices in IAEs/DAEs, STM-based trust-region scaling in trajectory optimization, and exponent determination in acoustic chain dynamics.

Searching arXiv for papers that explicitly use or define “Nonlinearity Index” across domains. I’m going to look up arXiv results for “nonlinearity index” and related variants to ground the article in the literature. Nonlinearity Index (NI) is not a single universal construct in the arXiv literature. The term is used in materially different ways across domains: as a harmonic-content diagnostic for nonlinear wave steepening in the sunspot umbral atmosphere, as a solution-dependent index notion for nonlinear integral-algebraic and differential-algebraic equations, as a local sensitivity measure for trust-region adaptation in successive convex programming, and as the exponent of a nonlinear interaction law in acoustic chains. Several neighboring literatures also discuss “nonlinearity” without defining a formal NI, and instead use surrogate quantities such as the nonlinear refractive index $n_2$, preservation of $d_{33}$, the energy dispersion index (EDI), extra-normal information $I_E$, or the number and strength of spline knots. Taken together, this suggests that NI is best understood as a domain-specific diagnostic rather than a standardized scalar observable (Sanjay et al., 14 Jul 2025, Shiri, 2013, Tafazzol et al., 11 Nov 2025, Ivanchenko, 2010, Fernández et al., 2019, Rambu et al., 2019, Wu et al., 2021, Hartman et al., 2018, Qu et al., 2024).

1. Cross-domain meanings and terminological scope

The arXiv record supports several technically distinct uses of NI. In solar physics, NI is explicitly defined from harmonic power. In nonlinear IAEs/DAEs, the nonlinear index is inherited from a linearized problem in a neighborhood of the exact solution. In low-thrust trajectory optimization, NI is an STM-based measure of local nonlinearity used to scale trust regions. In lattice dynamics, “nonlinearity index” denotes the exponent $\gamma$ in the nonlinear force term. A separate source of ambiguity is that in control theory the abbreviation NI commonly means “negative imaginary,” not “nonlinearity index” (Sanjay et al., 14 Jul 2025, Shiri, 2013, Tafazzol et al., 11 Nov 2025, Ivanchenko, 2010, Ghallab et al., 2022).

Context	NI meaning or closest quantity	Paper
Sunspot slow magnetoacoustic waves	$NI=\sqrt{P_2/P_1}$ from wavelet harmonic power	(Sanjay et al., 14 Jul 2025)
Nonlinear IAEs/DAEs	Index of the associated linearized equation near the exact solution	(Shiri, 2013)
SCP for low-thrust trajectories	STM-based local nonlinearity metric $v(\tau,\tau_0)$	(Tafazzol et al., 11 Nov 2025)
Acoustic chains	Arbitrary nonlinearity index $\gamma$ in the interaction potential	(Ivanchenko, 2010)
Nonlinear systems theory	NI denotes “negative imaginary,” not a nonlinearity index	(Ghallab et al., 2022)

This terminological dispersion matters methodologically. A harmonic-ratio NI, a tractability index, an STM-sensitivity ratio, and a power-law exponent are not interchangeable. Their mathematical content, identifiability, and physical interpretation are determined by the surrounding model class rather than by the label alone.

2. Harmonic-ratio NI in sunspot wave observations

In the study of 3-minute slow magnetoacoustic waves in the sunspot umbral atmosphere, NI is an observational measure of departure from a purely sinusoidal, linear waveform. The working premise is that wave steepening transfers power from the fundamental into higher harmonics, especially the second harmonic. The paper therefore defines

$NI=\sqrt{\frac{P_2}{P_1}},$

with $P_1$ the power in the fundamental mode and $P_2$ the power in the second harmonic. In amplitude language this is $d_{33}$0, so small NI corresponds to a nearly linear wave and large NI to stronger distortion or stronger nonlinearity (Sanjay et al., 14 Jul 2025).

The computation is pixel-by-pixel over the umbral region. The procedure uses HMI continuum intensity for umbral segmentation, subtraction of a 10-minute running average for detrending, and a complex Morlet wavelet following Torrence and Compo (1998). The dominant period in each channel is determined from histograms with 20-second bins, and a representative fundamental period $d_{33}$1 is selected for each active region. Fundamental and second-harmonic bands are then defined as $d_{33}$2 and the same relative width around half the fundamental period. Wavelet power is summed over these bands to obtain $d_{33}$3 and $d_{33}$4. A threshold based on the time-averaged value of the fundamental signal suppresses spurious ratios from weak signals; if the fundamental component is below threshold, the corresponding pixel is assigned $d_{33}$5. Channel-level NI is obtained by fitting a Gaussian curve to the histogram of nonzero NI values and taking the peak as the representative NI.

Across 20 active regions, the reported vertical pattern is systematic: HMI continuum shows very low NI, AIA 1600 Å rises, AIA 1700 Å reaches the maximum, AIA 304 Å decreases, AIA 131 Å rises again, and AIA 171 Å declines again. The authors interpret this as two phases of nonlinear evolution: one between the AIA 1700 Å and AIA 304 Å formation layers, and another between AIA 131 Å and AIA 171 Å. They connect the first phase to progressive steepening and shock formation in the lower atmosphere, consistent with median formation heights around 368 km for AIA 1700 Å and 858 km for AIA 304 Å. The paper also notes practical caveats: the HMI continuum harmonic band lies close to the Nyquist limit, large NI values can be produced by weak fundamentals or misassigned period bands, and cadence limitations prevent reliable inclusion of higher-order harmonics.

3. Solution-dependent nonlinear index for IAEs and DAEs

For nonlinear differential-algebraic equations and integral-algebraic equations, the relevant NI is not a harmonic ratio but a generalization of the linear rank degree or tractable index. The systems studied are

$d_{33}$6

and

$d_{33}$7

with singular $d_{33}$8 of constant rank. The central point is that, unlike the linear case, the nonlinear index depends on the exact solution. The paper therefore defines the nonlinear index by linearizing around the exact solution through the mean value theorem, computing the index of the associated linearized equation in a neighborhood of that solution, and declaring that index to be the nonlinear index (Shiri, 2013).

This construction is carried out through the residual equations for an approximate solution $d_{33}$9, with error $I_E$0. After linearization one obtains

$I_E$1

for the DAE case, and

$I_E$2

for the IAE case, where $I_E$3 lies in a neighborhood of the exact solution. The paper’s formal definition is: the index of the nonlinear equations is $I_E$4 if there exists a neighborhood $I_E$5 of the exact solution $I_E$6 in which the associated linearized equation has index $I_E$7.

Because $I_E$8 and $I_E$9 can change rank or structure with the solution, the same nonlinear system may have one index on one region of the solution and another near a critical point. This motivates the paper’s classification into well structure and free structure. A nonlinear IAE/DAE is well structure if its index does not depend on the unknown variables. It is free structure if its index does depend on the unknown variables; this is subdivided into dependent form, where the index changes along the exact solution, and independent form, where the index remains constant on the interval of interest. Critical conditions are conditions that change the index in a free-structure problem, and critical points are points where the exact solution satisfies such a condition.

The numerical implication is explicit. The paper warns that standard numerical methods such as Runge–Kutta methods, multistep methods, and collocation methods can fail after the first critical point if the index changes along the solution. Its examples show that a solver may behave well while the solution remains in a region of constant index and then deviate badly once a critical condition is encountered. In this literature, NI is therefore not a single formula attached to coefficients alone; it is a local structural property of the exact solution branch.

4. STM-based NI in sequential convex optimization for low-thrust trajectories

In adaptive-mesh successive convex programming for minimum-fuel low-thrust trajectory design, NI is introduced as a local measure of how nonlinear the trajectory dynamics are over a segment. The motivation is that SCVX relies on local linear approximations, and linearization error grows where the dynamics are more nonlinear. The paper therefore compares the state transition matrix $\gamma$0 for a nominal initial state with perturbed STMs $\gamma$1, and defines a nonlinearity index $\gamma$2 from their normalized deviation. It also develops a tensor-approximated NI using the second-order state transition tensor $\gamma$3, together with a directional NI $\gamma$4 for perturbations in a single state direction (Tafazzol et al., 11 Nov 2025).

The operational rule is simple: large NI implies a smaller trust region, and small NI implies a larger trust region. NI therefore acts as a local trust-region gain, complementing the standard ratio-based SCVX update that uses $\gamma$5 to accept or reject a step and expand or shrink the trust region. The broader framework combines time-dilation variables $\gamma$6 for adaptive mesh refinement, ratio-based global trust-region management, and segment-wise NI-based local trust-region scaling. The paper emphasizes that mesh refinement improves resolution, whereas NI-based trust regions improve convexification validity.

Two benchmark problems illustrate the effect. In the Earth-to-Dionysus case, the method converged in 38 iterations with $\gamma$7 and final mass $\gamma$8 kg, close to the known indirect-method optimum of $\gamma$9 kg. The authors report that mesh refinement did not materially improve the objective and that NI reduced iterations only marginally, plausibly because modified equinoctial elements already reduce nonlinearity. In the Earth–Moon $NI=\sqrt{P_2/P_1}$0 Halo-to-Halo CR3BP case, the method converged in 25 iterations with $NI=\sqrt{P_2/P_1}$1, final mass $NI=\sqrt{P_2/P_1}$2 kg, propellant consumption $NI=\sqrt{P_2/P_1}$3 kg, and transfer time $NI=\sqrt{P_2/P_1}$4 days. At lower discretization levels, mesh refinement produced a noticeably higher final mass than the uniform-mesh case, and the NI heuristic consistently reduced iteration count. The reported trust-region behavior shrinks near the Moon, where nonlinearity is stronger, and expands away from it. A plausible implication is that this NI is most useful when the underlying dynamics are strongly nonlinear rather than merely weakly nonlinear.

5. Nonlinearity index as exponent in acoustic-chain dynamics

In FPU-type acoustic chains, “arbitrary nonlinearity index” refers to the exponent $NI=\sqrt{P_2/P_1}$5 of the nonlinear interaction term, not to a post hoc scalar diagnostic. The equations of motion are

$NI=\sqrt{P_2/P_1}$6

so $NI=\sqrt{P_2/P_1}$7 is the power-law exponent of the nonlinear force term. The familiar $NI=\sqrt{P_2/P_1}$8-FPU and $NI=\sqrt{P_2/P_1}$9-FPU cases are recovered at $v(\tau,\tau_0)$0 and $v(\tau,\tau_0)$1, respectively (Ivanchenko, 2010).

The paper studies q-breathers, i.e. exact time-periodic continuations of linear normal modes that remain exponentially localized in mode space, and shows that the qualitative behavior of localization, stability, and thermalization is governed by a critical index

$v(\tau,\tau_0)$2

The decisive scaling comes from a localization factor containing $v(\tau,\tau_0)$3. For $v(\tau,\tau_0)$4, both delocalization and stability thresholds in nonlinearity tend to zero with increasing system size. For $v(\tau,\tau_0)$5, the leading-order behavior is size-independent. For $v(\tau,\tau_0)$6, both thresholds diverge with system size, so larger chains strengthen localization and suppress thermalization.

The same crossover appears in the Floquet-type stability analysis and in direct numerical tests of thermalization using the participation number

$v(\tau,\tau_0)$7

For $v(\tau,\tau_0)$8, $v(\tau,\tau_0)$9 grows rapidly, indicating equipartition. For $\gamma$0, $\gamma$1 remains small on the observed time scale. The paper concludes that, for a generic interaction potential, mode-space localized dynamics is determined only by the three lowest-order nonlinear terms in the power-series expansion. In this literature, NI is thus a constitutive exponent with direct consequences for localization, instability, weak chaos, and thermalization.

6. Proxy metrics, adjacent constructions, and cases with no formal NI

A substantial portion of the literature discusses nonlinearity without defining a formal NI. In stellar-population spectroscopy of M31 globular clusters, the relevant mechanism is the nonlinear metallicity-to-index conversion, especially the hot horizontal-branch contribution to Balmer lines and Mg$\gamma$2. The paper explicitly states that it does not define a formal quantity called a “Nonlinearity Index”; instead, nonlinearity is assessed from the shape of the $\gamma$3-index relations, the presence of quasi-inflection points, and the resulting projection of a unimodal metallicity distribution into bimodal index histograms (Kim et al., 2013).

In nonlinear optics, the “Method for real-time measurement of the nonlinear refractive index” is explicitly about continuous real-time measurement of $\gamma$4, not a separate NI. The operative nonlinearity measure is the nonlinear refractive index $\gamma$5, or equivalently the SPM-induced nonlinear phase shift $\gamma$6, inferred from spectral broadening and dispersive Fourier transformation. The method is non-interferometric, non-scanning, single-shot capable, and demonstrated at 30 MHz, but the paper states that it does not introduce a separate quantity literally named NI (Fernández et al., 2019). In lithium niobate waveguides fabricated by HiVac-VPE, the closest proxy is preservation of the second-order nonlinear coefficient $\gamma$7, assessed by SHG response. The authors conclude that the nonlinear coefficient is at least equal to the substrate value and that no measurable degradation occurs, while also stating that the exact $\gamma$8 cannot be extracted from the experiment (Rambu et al., 2019).

In coherent optical transmission, the closest analogue to NI is the energy dispersion index. The L-CCDM paper states that it does not introduce a separate NI formula; instead, EDI is the operative nonlinearity-tolerance metric. EDI is defined from a moving-window energy process, and the paper’s interpretive claim is that, with a proper window length, the lower the EDI of transmitted symbols, the less nonlinear interference is induced. L-CCDM exploits multiple candidate constant-composition sequences and selects the one with the smallest EDI, producing reported gains of 0.35 dB effective SNR, 0.22 bit/4D-symbol achievable information rate, or an 8% reach extension relative to standard CCDM with 256QAM (Wu et al., 2021).

In stock-network analysis, there is likewise no named NI. The closest pairwise scalar is the extra-normal information

$\gamma$9

which measures how much the observed dependence exceeds the Gaussian mutual information implied by the same correlation. The paper uses marginal normalization, surrogate-data tests, and row/column sums of $NI=\sqrt{\frac{P_2}{P_1}},$0 to separate genuine nonlinear coupling from univariate non-Gaussianity and nonstationarity. Its main conclusion is that the apparent nonlinearity observed in NYSE100, FTSE100, and SP500 stock networks is largely due to univariate non-Gaussianity, with additional contribution from strong nonstationarity during the 2008 financial crisis (Hartman et al., 2018).

In linear spline index regression, the paper again states that no explicit NI is defined. Nonlinearity is represented by the presence, number, and estimated locations of knots in the single-index effect, by the knot coefficients $NI=\sqrt{\frac{P_2}{P_1}},$1 that measure slope changes, and by a formal hypothesis test

$NI=\sqrt{\frac{P_2}{P_1}},$2

The estimated number of knots $NI=\sqrt{\frac{P_2}{P_1}},$3, the magnitudes of the $NI=\sqrt{\frac{P_2}{P_1}},$4, and the supremum test statistic $NI=\sqrt{\frac{P_2}{P_1}},$5 are therefore the closest surrogates for a nonlinearity measure in that framework (Qu et al., 2024).

These adjacent constructions reinforce a general point. The phrase “Nonlinearity Index” sometimes denotes a specific formal quantity, but in many technically mature literatures the same role is played by a domain-native object: a harmonic ratio, a Jacobian-dependent tractability index, an STM-sensitivity ratio, a nonlinear refractive index, an SHG-preservation criterion, an energy-dispersion statistic, an excess mutual information, or knot structure. The term is therefore best interpreted through the model class and measurement pipeline in which it is embedded.