Phase-Dependent Amplitude in Complex Systems

• Phase-Dependent Amplitude (PDA) is a framework where amplitude systematically varies with phase, impacting oscillatory, wave, neural, and RIS systems.
• The analysis uses small-parameter expansions and near-identity transformations to reveal higher-order, nonpairwise interactions and modified stability properties.
• PDA integrates phase-dependent modulation across diverse applications, emphasizing the need for accurate coupling models to predict dynamics and control performance.

Searching arXiv for the cited PDA-related works to ground the article in current metadata. Phase-Dependent Amplitude (PDA) denotes a class of phase–amplitude couplings in which amplitude is not independent of phase, but varies systematically with it. Across the literature, the term is used in several technically distinct settings: in nonlinear oscillators, PDA means that the norm of the state along a stable limit cycle depends on the phase, so that the limit-cycle embedding is noncircular (Bick et al., 2023); in wave dynamics on finite background, PDA refers to an explicit phase dependence of the effective potential governing amplitude evolution (Groesen et al., 2019); in reconfigurable intelligent surfaces (RISs), PDA denotes the deterministic dependence of reflection amplitude on programmed phase (Ozturk et al., 2022, Yilmaz et al., 28 Apr 2026); and in phase–amplitude models of neural dynamics, PDA describes amplitude dynamics modulated by phase differences or phase-dependent response functions (Fagerholm et al., 2019, Nicks et al., 2024). Despite these differences, the unifying feature is that amplitude cannot be treated as a passive or constant quantity once phase is specified.

1. Terminological scope and core definitions

In the oscillator-network formulation of "Higher-Order Network Interactions through Phase Reduction for Oscillators with Phase-Dependent Amplitude" (Bick et al., 2023), PDA is defined by the requirement that the norm of the state on a stable limit cycle depends on phase. In polar coordinates $A = r e^{i\phi}$, the limit cycle is parameterized by

$r_0(\phi) = 1 + \delta\, g(\phi),$

with $g$ a smooth $2\pi$-periodic function and $0 < |\delta| \ll 1$. The corresponding embedding in $\mathbb{C} \cong \mathbb{R}^2$ is

$X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$

The circular Stuart–Landau case corresponds to $g \equiv 0$, hence $r_0(\phi)\equiv 1$, and therefore has no PDA (Bick et al., 2023).

The same paper realizes PDA through the uncoupled single-oscillator dynamics

$$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$

where $r_0(\phi) = 1 + \delta\, g(\phi),$0 is the angular velocity and $r_0(\phi) = 1 + \delta\, g(\phi),$1 the radial attraction rate. The prescribed limit cycle $r_0(\phi) = 1 + \delta\, g(\phi),$2 is invariant and asymptotically stable in $r_0(\phi) = 1 + \delta\, g(\phi),$3 (Bick et al., 2023).

In the nonlinear Schrödinger finite-background setting of "Displaced phase-amplitude variables for waves on finite background" (Groesen et al., 2019), PDA appears in a different but related sense: the Hamiltonian potential energy depends explicitly on phase through

$r_0(\phi) = 1 + \delta\, g(\phi),$4

For time-independent displaced phase $r_0(\phi) = 1 + \delta\, g(\phi),$5, the amplitude variable $r_0(\phi) = 1 + \delta\, g(\phi),$6 obeys an autonomous oscillator equation whose coefficients depend on $r_0(\phi) = 1 + \delta\, g(\phi),$7, so amplitude evolution is driven by phase and by the spatial phase gradient $r_0(\phi) = 1 + \delta\, g(\phi),$8 (Groesen et al., 2019).

In RIS hardware models, PDA denotes phase-dependent amplitude response of an individual element. In "RIS-aided Near-Field Localization under Phase-Dependent Amplitude Variations" (Ozturk et al., 2022), the element response is

$r_0(\phi) = 1 + \delta\, g(\phi),$9

with

$g$0

where $g$1. In "Performance Analysis of HAPS-RIS-Assisted MIMO Systems Under Phase-Dependent Amplitude Response Using Saddle Point Approximation" (Yilmaz et al., 28 Apr 2026), the RIS element model is

$g$2

with discrete phase shifts $g$3 (Yilmaz et al., 28 Apr 2026).

A broader conceptual use appears in dynamic causal modelling of neural systems, where amplitude change depends on phase differences. In "Dynamic causal modelling of phase-amplitude interactions" (Fagerholm et al., 2019), amplitude coupling elements $g$4 are multiplied by $g$5, so that amplitude dynamics are explicitly phase dependent.

2. PDA in phase reduction of limit-cycle oscillators

The principal oscillator-theoretic contribution of (Bick et al., 2023) is the extension of second-order phase reduction beyond circular limit cycles to arbitrary networks of coupled nonlinear oscillators with PDA. The starting point is the weakly coupled network

$g$6

with smooth $g$7, weak coupling $g$8, and stable limit cycles for uncoupled units (Bick et al., 2023). The specific mean-field model studied is

$g$9

and its graph-coupled generalization is

$2\pi$0

where $2\pi$1 is the adjacency matrix of a possibly directed or weighted coupling graph $2\pi$2 (Bick et al., 2023).

A key technical step is the near-identity transformation

$2\pi$3

which removes the phase-dependent radius and maps the invariant torus to a circular one at $2\pi$4 when $2\pi$5. The resulting phase–radius system is

$2\pi$6

with explicit $2\pi$7 given in terms of $2\pi$8, $2\pi$9, and the phase differences (Bick et al., 2023).

For the PDA model with $0 < |\delta| \ll 1$0, the asymptotic phase map is particularly simple: $0 < |\delta| \ll 1$1 so the isochrons are $0 < |\delta| \ll 1$2, and the infinitesimal phase response curve is

$0 < |\delta| \ll 1$3

This simplicity is specific to the chosen PDA construction; the paper contrasts it with the Stuart–Landau case having rotational symmetry and $0 < |\delta| \ll 1$4 (Bick et al., 2023).

The first-order reduced phase dynamics on the invariant torus is

$0 < |\delta| \ll 1$5

Expanding in $0 < |\delta| \ll 1$6 yields

$0 < |\delta| \ll 1$7

with

$0 < |\delta| \ll 1$8

$0 < |\delta| \ll 1$9

$\mathbb{C} \cong \mathbb{R}^2$0

Thus PDA modifies pairwise phase coupling already at first order (Bick et al., 2023).

At second order, the general structure becomes

$\mathbb{C} \cong \mathbb{R}^2$1

where

$\mathbb{C} \cong \mathbb{R}^2$2

The first torus deformation $\mathbb{C} \cong \mathbb{R}^2$3 is determined from the linear first-order PDE

$\mathbb{C} \cong \mathbb{R}^2$4

with $\mathbb{C} \cong \mathbb{R}^2$5. This PDE is solved order-by-order in $\mathbb{C} \cong \mathbb{R}^2$6, with explicit formulas for $\mathbb{C} \cong \mathbb{R}^2$7, and, for $\mathbb{C} \cong \mathbb{R}^2$8, explicit $\mathbb{C} \cong \mathbb{R}^2$9 and $X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$0 (Bick et al., 2023).

A central structural consequence is that PDA breaks rotational $X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$1 symmetry: $X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$2 depend explicitly on $X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$3, so one cannot set $X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$4 by co-rotation. This absence of a co-rotating-frame simplification affects both derivation and stability analysis (Bick et al., 2023).

3. Higher-order interactions, hypergraphs, and network structure

At second order, the reduction in (Bick et al., 2023) generates genuine nonpairwise interactions. For all-to-all coupling and $X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$5, the second-order term contains triplet interactions of the form

$X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$6

which couples the three phases $X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$7 (Bick et al., 2023). This is not a phenomenological addition; it emerges from additive pairwise coupling between limit-cycle oscillators once second-order corrections are retained.

For graph coupling, the first-order reduction is

$X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$8

while the second-order $X_0(\phi) = r_0(\phi)(\cos\phi,\sin\phi).$9 term decomposes into products of adjacency entries: $g \equiv 0$0 with kernels

$g \equiv 0$1

$g \equiv 0$2

The paper then introduces directed 3-uniform hyperedge tensors

$g \equiv 0$3

so that the second-order term becomes a sum over hyperedges (Bick et al., 2023).

This construction yields a natural hypergraph interpretation of second-order phase reduction. The identified classes of interactions include pairwise corrections along the original graph, pairwise terms along “virtual” edges associated with paths $g \equiv 0$4, and triplet hyperedges of two types. Even if the underlying graph is undirected, the induced hypergraph is often directed unless the graph is all-to-all (Bick et al., 2023). A plausible implication is that directed higher-order interactions may arise generically in reduced descriptions even when the microscopic coupling law itself is pairwise and symmetric.

The paper emphasizes that PDA adds further second-order terms $g \equiv 0$5 and $g \equiv 0$6 through three mechanisms: the explicit $g \equiv 0$7-dependence of $g \equiv 0$8, the appearance of $g \equiv 0$9 and $r_0(\phi)\equiv 1$0, and the deformation fields $r_0(\phi)\equiv 1$1 and $r_0(\phi)\equiv 1$2, which depend on $r_0(\phi)\equiv 1$3 and its derivatives (Bick et al., 2023). Consequently, PDA affects not only the strength of higher-order interactions but also their detailed harmonic content.

4. Stability, bifurcations, and dynamical consequences

In the full system with $r_0(\phi)\equiv 1$4, synchrony admits a linearization whose critical Floquet exponents are

$r_0(\phi)\equiv 1$5

together with the neutral exponent $r_0(\phi)\equiv 1$6. The corresponding Floquet multipliers are $r_0(\phi)\equiv 1$7, with $r_0(\phi)\equiv 1$8 (Bick et al., 2023).

For the phase-reduced systems, because co-rotation is unavailable when $r_0(\phi)\equiv 1$9, the stability analysis uses Floquet multipliers of the time-dependent linearization. For the first-order reduction, the critical multiplier is

$$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$0

which is independent of $$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$1. For the second-order $$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$2 reduction,

$$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$3

For the $$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$4 truncation with $$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$5,

$$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$6

which exhibits the leading $$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$7 correction (Bick et al., 2023). This shows that PDA can influence synchrony stability only at even order in $$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$8 in the analyzed setting.

For splay states, the paper defines the splay manifold

$$\dot r = \delta g'(\phi)\,\omega\,\frac{r}{1+\delta g(\phi)} + m\,r^2\,(r - 1 - \delta g(\phi)),\qquad \dot \phi = \omega,$$9

When $r_0(\phi) = 1 + \delta\, g(\phi),$00, both the full and reduced systems admit the periodic orbit

$r_0(\phi) = 1 + \delta\, g(\phi),$01

For $r_0(\phi) = 1 + \delta\, g(\phi),$02, the linearization yields in the $r_0(\phi) = 1 + \delta\, g(\phi),$03 reduction

$r_0(\phi) = 1 + \delta\, g(\phi),$04

and in the $r_0(\phi) = 1 + \delta\, g(\phi),$05 reduction

$r_0(\phi) = 1 + \delta\, g(\phi),$06

Numerical analysis for $r_0(\phi) = 1 + \delta\, g(\phi),$07 shows a subcritical Neimark–Sacker bifurcation in the full system that is captured by the $r_0(\phi) = 1 + \delta\, g(\phi),$08 reduction but missed by the first-order reduction (Bick et al., 2023). This directly supports the paper’s claim that first-order truncations can mispredict stability and miss bifurcations when the limit cycle is noncircular or the coupling is stronger.

The practical implication drawn in (Bick et al., 2023) is that PDA modifies both pairwise coupling and the emergence of nonpairwise interactions, thereby affecting synchronization thresholds, phase-locking regions, and collective dynamics.

5. PDA beyond oscillator networks: waves, neural systems, and delay equations

In the displaced phase–amplitude description of the focusing nonlinear Schrödinger equation on a finite monochromatic background (Groesen et al., 2019), the field is written as

$r_0(\phi) = 1 + \delta\, g(\phi),$09

and the transformed Hamiltonian contains the phase-dependent potential

$r_0(\phi) = 1 + \delta\, g(\phi),$10

For the special class $r_0(\phi) = 1 + \delta\, g(\phi),$11, the amplitude $r_0(\phi) = 1 + \delta\, g(\phi),$12 satisfies

$r_0(\phi) = 1 + \delta\, g(\phi),$13

or equivalently

$r_0(\phi) = 1 + \delta\, g(\phi),$14

where $r_0(\phi) = 1 + \delta\, g(\phi),$15 acts as a parameter in the effective potential (Groesen et al., 2019). In this setting, the paper explicitly states that the change of phase with position is the only driving force in the autonomous oscillator for $r_0(\phi) = 1 + \delta\, g(\phi),$16, and uses this PDA mechanism to interpret the Akhemediev breather, Ma breather, and Peregrine soliton as members of the same constrained phase-parameterized family.

In dynamic causal modelling of neural systems (Fagerholm et al., 2019), PDA is operationalized directly in the state equations. The phase dynamics is

$r_0(\phi) = 1 + \delta\, g(\phi),$17

while the amplitude dynamics is

$r_0(\phi) = 1 + \delta\, g(\phi),$18

Here the factors $r_0(\phi) = 1 + \delta\, g(\phi),$19 implement explicit phase-dependent amplitude coupling (Fagerholm et al., 2019). The paper reports that phase-only models perform well only under weak coupling conditions, whereas phase–amplitude models describe strongly coupled systems more effectively, and capture the Kuramoto order parameter, cross-correlation, and phase-lag index more effectively than phase-only models (Fagerholm et al., 2019). This suggests that PDA becomes empirically relevant when the system explores states away from a strongly attracting limit cycle.

For delay-differential equations, "Phase and amplitude responses for delay equations using harmonic balance" (Nicks et al., 2024) frames PDA through the reduced system

$r_0(\phi) = 1 + \delta\, g(\phi),$20

where $r_0(\phi) = 1 + \delta\, g(\phi),$21 is the phase-dependent amplitude response function. The associated adjoint DDE for the amplitude response $r_0(\phi) = 1 + \delta\, g(\phi),$22 is

$r_0(\phi) = 1 + \delta\, g(\phi),$23

with normalization

$r_0(\phi) = 1 + \delta\, g(\phi),$24

The reduced amplitude equation makes PDA explicit: the same forcing $r_0(\phi) = 1 + \delta\, g(\phi),$25 produces different amplitude increments depending on phase through $r_0(\phi) = 1 + \delta\, g(\phi),$26 (Nicks et al., 2024).

6. PDA in reconfigurable intelligent surfaces and communication systems

In RIS research, PDA is a hardware impairment model rather than a dynamical invariant-manifold phenomenon. Each reflecting element has a complex coefficient whose amplitude depends on the selected phase. In near-field localization (Ozturk et al., 2022), the received signal through an RIS is

$r_0(\phi) = 1 + \delta\, g(\phi),$27

with

$r_0(\phi) = 1 + \delta\, g(\phi),$28

If the receiver assumes a unit-amplitude RIS response $r_0(\phi) = 1 + \delta\, g(\phi),$29, model mismatch arises (Ozturk et al., 2022).

The paper derives a misspecified Cramér–Rao bound and a lower bound on mean-squared error, and reports severe performance penalties when PDA is ignored, especially at high signal-to-noise ratio. For example, at $r_0(\phi) = 1 + \delta\, g(\phi),$30 dB SNR in the reported setup, the lower bound grows from approximately $r_0(\phi) = 1 + \delta\, g(\phi),$31 m at $r_0(\phi) = 1 + \delta\, g(\phi),$32 to approximately $r_0(\phi) = 1 + \delta\, g(\phi),$33 m at $r_0(\phi) = 1 + \delta\, g(\phi),$34 and approximately $r_0(\phi) = 1 + \delta\, g(\phi),$35–$r_0(\phi) = 1 + \delta\, g(\phi),$36 m as $r_0(\phi) = 1 + \delta\, g(\phi),$37, whereas the correctly specified CRB remains approximately $r_0(\phi) = 1 + \delta\, g(\phi),$38–$r_0(\phi) = 1 + \delta\, g(\phi),$39 m (Ozturk et al., 2022). The paper also states that ignoring PDA can require more than $r_0(\phi) = 1 + \delta\, g(\phi),$40 more elements to match the accuracy obtained with a correctly modeled RIS (Ozturk et al., 2022).

To mitigate this, the paper proposes joint estimation of user location and RIS amplitude model parameters, with a joint Fisher information matrix for

$r_0(\phi) = 1 + \delta\, g(\phi),$41

and an alternating refinement algorithm that calibrates $r_0(\phi) = 1 + \delta\, g(\phi),$42, $r_0(\phi) = 1 + \delta\, g(\phi),$43, and $r_0(\phi) = 1 + \delta\, g(\phi),$44 online (Ozturk et al., 2022). The reported simulation results indicate fast convergence and performance close to the CRB (Ozturk et al., 2022).

In HAPS-RIS-assisted MIMO systems (Yilmaz et al., 28 Apr 2026), PDA enters the effective channel through two aggregated quantities,

$r_0(\phi) = 1 + \delta\, g(\phi),$45

Under a central-limit approximation, the cascaded channel entries are approximately Gaussian with mean

$r_0(\phi) = 1 + \delta\, g(\phi),$46

and variance

$r_0(\phi) = 1 + \delta\, g(\phi),$47

The post-MRC SNR is modeled as a non-central quadratic form

$r_0(\phi) = 1 + \delta\, g(\phi),$48

with $r_0(\phi) = 1 + \delta\, g(\phi),$49 and

$r_0(\phi) = 1 + \delta\, g(\phi),$50

PDA therefore affects both the non-centrality and covariance of the SNR statistic (Yilmaz et al., 28 Apr 2026).

The paper derives a saddle-point-approximation framework for the SNR distribution. The cumulant generating function is

$r_0(\phi) = 1 + \delta\, g(\phi),$51

with corresponding SPA expressions for the PDF, CDF, and outage probability (Yilmaz et al., 28 Apr 2026). It reports that PDA reduces coherent gain and increases outage, while higher phase resolution mitigates the penalty (Yilmaz et al., 28 Apr 2026).

A related beamforming problem is studied in "Discrete Beamforming Optimization for RISs with a Limited Phase Range and Amplitude Attenuation" (Pekcan et al., 9 Jul 2025), where the RIS coefficient is

$r_0(\phi) = 1 + \delta\, g(\phi),$52

and the practical PDA curve is

$r_0(\phi) = 1 + \delta\, g(\phi),$53

The paper derives necessary and sufficient optimality conditions for the discrete beamforming problem, and an optimal search algorithm that converges in linear time within at most $r_0(\phi) = 1 + \delta\, g(\phi),$54 steps (Pekcan et al., 9 Jul 2025). It further proposes amplitude-introduced polar quantization (APQ) and extended amplitude-introduced polar quantization (EAPQ), and reports that increasing the number of discrete phases beyond $r_0(\phi) = 1 + \delta\, g(\phi),$55 yields only marginal gains, provided the phase range $r_0(\phi) = 1 + \delta\, g(\phi),$56 is sufficiently wide (Pekcan et al., 9 Jul 2025).

7. Methods, assumptions, and limitations

Across these literatures, PDA is technically tractable only under fairly specific assumptions. In oscillator phase reduction (Bick et al., 2023), the analysis assumes weak coupling $r_0(\phi) = 1 + \delta\, g(\phi),$57, small deformation $r_0(\phi) = 1 + \delta\, g(\phi),$58, smooth $r_0(\phi) = 1 + \delta\, g(\phi),$59, persistence of a normally hyperbolic invariant $r_0(\phi) = 1 + \delta\, g(\phi),$60-torus, and sufficiently strong radial attraction $r_0(\phi) = 1 + \delta\, g(\phi),$61 to justify timescale separation. The paper explicitly notes that algebraic complexity becomes prohibitive beyond second order, that nonlinear coupling would complicate the torus PDE, that heterogeneity in $r_0(\phi) = 1 + \delta\, g(\phi),$62 reduces tractability, and that large $r_0(\phi) = 1 + \delta\, g(\phi),$63 would invalidate the small-parameter expansion (Bick et al., 2023).

In the finite-background NLS setting (Groesen et al., 2019), the PDA mechanism is developed for integrable focusing NLS in a space-evolution framework, and the autonomous oscillator reduction is tied to the special condition $r_0(\phi) = 1 + \delta\, g(\phi),$64. The paper notes that more general backgrounds may require time-dependent displaced phase $r_0(\phi) = 1 + \delta\, g(\phi),$65, rendering the oscillator non-autonomous (Groesen et al., 2019).

In neural phase–amplitude DCM (Fagerholm et al., 2019), the model assumes operation near a supercritical Hopf bifurcation with circular limit cycles and specific weighting choices $r_0(\phi) = 1 + \delta\, g(\phi),$66 and $r_0(\phi) = 1 + \delta\, g(\phi),$67. The paper notes that amplitude parameters can be less identifiable than phase parameters, and that under weak coupling the model reduces to the phase-only Kuramoto form, limiting identifiability of amplitude contributions (Fagerholm et al., 2019).

In delay equations (Nicks et al., 2024), the reduction retains only the leading isostable and is first-order in forcing amplitude. The authors note that multiple slow modes, complex Floquet exponents, and higher-order corrections remain to be incorporated (Nicks et al., 2024).

In RIS applications (Ozturk et al., 2022, Yilmaz et al., 28 Apr 2026, Pekcan et al., 9 Jul 2025), PDA models are hardware calibrated and effective rather than fundamental. Their practical success depends on the fidelity of the amplitude–phase curve $r_0(\phi) = 1 + \delta\, g(\phi),$68 or $r_0(\phi) = 1 + \delta\, g(\phi),$69, the quality of channel-state information, and the validity of simplifying assumptions such as narrowband operation, independence of angle and frequency in the RIS response, and tractable discrete codebooks. A plausible implication is that PDA in RIS systems is not merely a nuisance parameter: it defines the feasible geometry of the control space, alters identifiability and optimization landscapes, and can become the dominant source of model mismatch if idealized unit-amplitude assumptions are retained.

Overall, PDA serves as a common name for several mathematically distinct mechanisms in which phase and amplitude are inseparable. In nonlinear oscillators, it deforms invariant sets and induces higher-order interactions (Bick et al., 2023). In dispersive wave theory, it embeds phase into the effective amplitude potential (Groesen et al., 2019). In neural state-space models and delay equations, it enters as phase-dependent amplitude coupling or response functions (Fagerholm et al., 2019, Nicks et al., 2024). In RIS systems, it formalizes hardware-imposed amplitude attenuation as a function of phase and thereby reshapes inference and beamforming problems (Ozturk et al., 2022, Yilmaz et al., 28 Apr 2026, Pekcan et al., 9 Jul 2025). The shared methodological lesson is that whenever amplitude depends on phase in a structurally explicit way, phase-only descriptions can become quantitatively inaccurate or qualitatively incomplete.