---
title: Phase-Dependent Amplitude in Complex Systems
url: https://www.emergentmind.com/topics/phase-dependent-amplitude-pda
type: topic
---

# Phase-Dependent Amplitude in Complex Systems

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 [2305.04277]; in wave dynamics on finite background, PDA refers to an explicit phase dependence of the effective potential governing amplitude evolution [1906.00959]; in reconfigurable intelligent surfaces (RISs), PDA denotes the deterministic dependence of reflection amplitude on programmed phase [2204.12783, 2604.25331]; and in phase–amplitude models of neural dynamics, PDA describes amplitude dynamics modulated by phase differences or phase-dependent response functions [1909.08509, 2404.17356]. 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" [2305.04277], 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 [2305.04277].

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 \(\omega>0\) is the angular velocity and \(m<0\) the radial attraction rate. The prescribed limit cycle \(r = 1 + \delta g(\phi)\) is invariant and asymptotically stable in \(r\) [2305.04277].

In the nonlinear Schrödinger finite-background setting of "Displaced phase-amplitude variables for waves on finite background" [1906.00959], PDA appears in a different but related sense: the Hamiltonian potential energy depends explicitly on phase through
\[
W = \frac{\alpha}{4} G^2 (G - 2 \cos \phi)^2.
\]
For time-independent displaced phase \(\phi=\phi(\xi)\), the amplitude variable \(G\) obeys an autonomous oscillator equation whose coefficients depend on \(\phi\), so amplitude evolution is driven by phase and by the spatial phase gradient \(\lambda(\phi)=\partial_\xi\phi\) [1906.00959].

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" [2204.12783], the element response is
\[
w_{t,m} = \beta(\theta_{t,m}; \eta) e^{j\theta_{t,m}},
\]
with
\[
\beta(\theta; \eta) = (1 - \beta_{\min}) \left[\frac{\sin(\theta - \phi)+1}{2}\right]^\kappa + \beta_{\min},
\]
where \(\eta=[\beta_{\min},\kappa,\phi]^T\). In "Performance Analysis of HAPS-RIS-Assisted MIMO Systems Under Phase-Dependent Amplitude Response Using Saddle Point Approximation" [2604.25331], the RIS element model is
\[
\Gamma_n = \beta_n e^{j\varphi_n}, \qquad
\beta_n = (1-\zeta_{\min})\left(\frac{|\sin(\varphi_n-c)|}{2}\right)^k + \zeta_{\min},
\]
with discrete phase shifts \(\varphi_n\in Q_\varphi\) [2604.25331].

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" [1909.08509], amplitude coupling elements \(a_{r,ij}\) are multiplied by \(\cos(\theta_i-\theta_j)\), so that amplitude dynamics are explicitly phase dependent.

## 2. PDA in phase reduction of limit-cycle oscillators

The principal oscillator-theoretic contribution of [2305.04277] 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
\[
\dot x_i = F(x_i) + \varepsilon \sum_{j=1}^N C_{ij}\,G(x_i,x_j),\qquad i=1,\dots,N,
\]
with smooth \(F,G\), weak coupling \(0<\varepsilon\ll1\), and stable limit cycles for uncoupled units [2305.04277]. The specific mean-field model studied is
\[
\dot A_k = \mathcal F(A_k) + K e^{i\alpha}(\bar A - A_k),\qquad
\bar A = \frac{1}{N}\sum_{j=1}^N A_j,
\]
and its graph-coupled generalization is
\[
\dot A_k = \mathcal F(A_k) + \frac{K}{N} e^{i\alpha}\sum_{l=1}^N a_{kl}(A_l - A_k),
\]
where \(A=(a_{kl})\) is the adjacency matrix of a possibly directed or weighted coupling graph \(\Gamma=(V,E)\) [2305.04277].

A key technical step is the near-identity transformation
\[
R_k := \frac{r_k}{1+\delta g(\phi_k)},
\]
which removes the phase-dependent radius and maps the invariant torus to a circular one at \(R_k\equiv 1\) when \(K=0\). The resulting phase–radius system is
\[
\dot R_k = F(R_k,\phi_k) + K\,G_k(R,\phi),\qquad
\dot \phi_k = \omega + K\,H_k(R,\phi),
\]
with explicit \(F,G_k,H_k\) given in terms of \(g\), \(g'\), and the phase differences [2305.04277].

For the PDA model with \(c_2=0\), the asymptotic phase map is particularly simple:
\[
\Theta(r,\phi)=\phi,
\]
so the isochrons are \(\phi=\text{const}\), and the infinitesimal phase response curve is
\[
Z(\theta)=\nabla_x\Theta\big|_{x=X_0(\theta)}.
\]
This simplicity is specific to the chosen PDA construction; the paper contrasts it with the Stuart–Landau case having rotational symmetry and \(\Theta=\phi-c_2\ln r\) [2305.04277].

The first-order reduced phase dynamics on the invariant torus is
\[
\dot \phi_k = \omega + K\frac{1}{N}\sum_{l=1}^N
\left[
\frac{1+\delta g(\phi_l)}{1+\delta g(\phi_k)}
\sin(\phi_l-\phi_k+\alpha)-\sin\alpha
\right].
\]
Expanding in \(\delta\) yields
\[
\dot \phi_k = \omega + K\Big(P_k^{(1,0)}(\phi) + \delta P_k^{(1,1)}(\phi) + \delta^2 P_k^{(1,2)}(\phi) + \dots\Big),
\]
with
\[
P_k^{(1,0)}(\phi) = \frac{1}{N}\sum_{l=1}^N\left[\sin(\phi_l-\phi_k+\alpha)-\sin\alpha\right],
\]
\[
P_k^{(1,1)}(\phi) = \frac{1}{N}\sum_{l=1}^N\big(g(\phi_l)-g(\phi_k)\big)\sin(\phi_l-\phi_k+\alpha),
\]
\[
P_k^{(1,2)}(\phi) = \frac{1}{N}\sum_{l=1}^N g(\phi_k)\big(g(\phi_k)-g(\phi_l)\big)\sin(\phi_l-\phi_k+\alpha).
\]
Thus PDA modifies pairwise phase coupling already at first order [2305.04277].

At second order, the general structure becomes
\[
\dot \phi_k = \omega + K\,P_k^{(1,\star)}(\phi) + K^2\,P_k^{(2,\star)}(\phi) + \mathcal O(K^3),
\]
where
\[
P_k^{(2,\star)}(\phi) = \nabla_R H_k(1,\phi)\cdot R^{(1,\star)}(\phi).
\]
The first torus deformation \(R^{(1,\star)}\) is determined from the linear first-order PDE
\[
F_R(1,\phi_k)\,R_k^{(1,\star)}(\phi) + G_k(1,\phi)
= \omega\,\nabla_\phi R_k^{(1,\star)}(\phi)\cdot \mathbbm{1},
\]
with \(F_R(1,\phi_k)=m(1+\delta g(\phi_k))^2\). This PDE is solved order-by-order in \(\delta\), with explicit formulas for \(R_k^{(1,0)}\), and, for \(g(\phi)=\sin\phi\), explicit \(R_k^{(1,1)}\) and \(R_k^{(1,2)}\) [2305.04277].

A central structural consequence is that PDA breaks rotational \(S^1\) symmetry: \(F,G,H\) depend explicitly on \(\phi\), so one cannot set \(\omega=0\) by co-rotation. This absence of a co-rotating-frame simplification affects both derivation and stability analysis [2305.04277].

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

At second order, the reduction in [2305.04277] generates genuine nonpairwise interactions. For all-to-all coupling and \(\delta=0\), the second-order term contains triplet interactions of the form
\[
P^{(2,0)}_k(\phi) = \frac{1}{2N^2 m}\sum_{l=1}^N \sum_{i=1}^N
\Big(
\sin(\phi_i+\phi_k-2\phi_l)
-\sin(\phi_i-\phi_k+2\alpha)
+\sin(\phi_i-2\phi_k+\phi_l+2\alpha)
\Big),
\]
which couples the three phases \((i,k,l)\) [2305.04277]. 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
\[
\dot \phi_k = \omega + \frac{K}{N}\sum_{l=1}^N a_{kl}
\left[
\frac{1+\delta g(\phi_l)}{1+\delta g(\phi_k)}
\sin(\phi_l-\phi_k+\alpha)-\sin\alpha
\right],
\]
while the second-order \(\delta=0\) term decomposes into products of adjacency entries:
\[
P_k^{(2,0)}(\phi) =
-\frac{1}{2N^2 m}\sum_{l,i} a_{kl}a_{ki}\,\hat g(\phi_k,\phi_l,\phi_i)
+\frac{1}{2N^2 m}\sum_{l,i} a_{kl}a_{li}\,\bar g(\phi_k,\phi_l,\phi_i),
\]
with kernels
\[
\hat g(\phi_k,\phi_l,\phi_i) =
2\cos\alpha\,\sin(\phi_l-\phi_k+\alpha)
+\sin(\phi_i-\phi_l)
-\sin(\phi_i-2\phi_k+\phi_l+2\alpha),
\]
\[
\bar g(\phi_k,\phi_l,\phi_i) =
2\cos\alpha\,\sin(\phi_l-\phi_k+\alpha)
-\sin(\phi_i-\phi_k+2\alpha)
+\sin(\phi_i+\phi_k-2\phi_l).
\]
The paper then introduces directed 3-uniform hyperedge tensors
\[
\hat h_{kli} := a_{kl}a_{ki},\qquad \bar h_{kli} := a_{kl}a_{li},
\]
so that the second-order term becomes a sum over hyperedges [2305.04277].

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 \(i\to l\to k\), 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 [2305.04277]. 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 \(P^{(2,1)}\) and \(P^{(2,2)}\) through three mechanisms: the explicit \(\phi\)-dependence of \(F_R(1,\phi_k)\), the appearance of \(\nabla_R H_k^{(-,1)}\) and \(\nabla_R H_k^{(-,2)}\), and the deformation fields \(R^{(1,1)}\) and \(R^{(1,2)}\), which depend on \(g\) and its derivatives [2305.04277]. 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 \(\delta=0\), synchrony admits a linearization whose critical Floquet exponents are
\[
q_{2,\dots,N} = \tfrac{1}{2}\left(m - 2K\cos\alpha \pm \sqrt{m^2 - 2K^2 + 2K^2\cos(2\alpha)}\right),
\]
together with the neutral exponent \(q_1=0\). The corresponding Floquet multipliers are \(\lambda = e^{Tq}\), with \(T=2\pi/\omega\) [2305.04277].

For the phase-reduced systems, because co-rotation is unavailable when \(\delta\neq 0\), the stability analysis uses Floquet multipliers of the time-dependent linearization. For the first-order reduction, the critical multiplier is
\[
\lambda^\mathrm{crit} = \exp\left(-\frac{2\pi K}{\omega}\cos\alpha\right),
\]
which is independent of \(\delta\). For the second-order \((2,0)\) reduction,
\[
\lambda^\mathrm{crit} = \exp\left(-\frac{2\pi K}{m\omega}(m\cos\alpha - K\sin^2\alpha)\right).
\]
For the \((2,2)\) truncation with \(g(\phi)=\sin\phi\),
\[
\lambda^\mathrm{crit} =
\exp\Bigg(
\frac{-2\pi K}{m\omega (m^2+\omega^2)}
\Big[
m(m^2+\omega^2)\cos\alpha
- K(m^2+\omega^2)\sin^2\alpha
- 2K m^2 \delta^2 \sin^2\alpha
\Big]
\Bigg),
\]
which exhibits the leading \(\delta^2\) correction [2305.04277]. This shows that PDA can influence synchrony stability only at even order in \(\delta\) in the analyzed setting.

For splay states, the paper defines the splay manifold
\[
D = \{\phi\in\mathbb S^N:\phi_{k+1}=\phi_k+2\pi/N\}.
\]
When \(\delta=0\), both the full and reduced systems admit the periodic orbit
\[
\phi_k(t) = \hat\omega\,t + \frac{2\pi k}{N},\qquad
\hat\omega = \omega - K\sin\alpha,\qquad
T=\frac{2\pi}{\hat\omega}.
\]
For \(N=3\), the linearization yields in the \((1,0)\) reduction
\[
q_{2,3} = \frac{K}{2} e^{\pm i\alpha},
\]
and in the \((2,0)\) reduction
\[
q_{2,3} = \frac{K}{2} e^{\pm i\alpha}\left(1 - \frac{K}{2m} e^{\pm i\alpha}\right).
\]
Numerical analysis for \(\delta\neq 0\) shows a subcritical Neimark–Sacker bifurcation in the full system that is captured by the \((2,2)\) reduction but missed by the first-order reduction [2305.04277]. 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 [2305.04277] 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 [1906.00959], the field is written as
\[
A(\xi,\tau) = (G(\xi,\tau)e^{i\phi(\xi,\tau)} - 1)\,r_0 e^{-i\alpha \xi},
\]
and the transformed Hamiltonian contains the phase-dependent potential
\[
W(G,\phi) = \frac{\alpha}{4}G^2(G-2\cos\phi)^2.
\]
For the special class \(\phi_\tau=0\), the amplitude \(G\) satisfies
\[
\phi_\xi G + \beta G_{\tau\tau} + \frac{\partial W}{\partial G}=0,
\]
or equivalently
\[
\partial_\tau^2 G + \lambda G + \alpha G(G-\cos\phi)(G-2\cos\phi)=0,
\]
where \(\lambda(\phi)=\partial_\xi\phi\) acts as a parameter in the effective potential [1906.00959]. In this setting, the paper explicitly states that the change of phase with position is the only driving force in the autonomous oscillator for \(G\), 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 [1909.08509], PDA is operationalized directly in the state equations. The phase dynamics is
\[
\frac{d\theta_i}{dt}
=
\Omega_i
+
\sum_j a_{e,ij} e^{-|r_i-r_j|}\sin(\theta_j-\theta_i)
-
\sum_k C_{ki}v_k \frac{\sin\theta_i}{r_i},
\]
while the amplitude dynamics is
\[
\frac{dr_i}{dt}
=
\left(1-\frac{r_{\mathrm{LC},i}}{r_i}\right)
\sum_j a_{r,ij}\cos(\theta_i-\theta_j) r_i
+
\sum_j v_j B_{ij} r_i
+
\sum_k C_{ki} v_k \cos\theta_i.
\]
Here the factors \(\cos(\theta_i-\theta_j)\) implement explicit phase-dependent amplitude coupling [1909.08509]. 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 [1909.08509]. 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" [2404.17356] frames PDA through the reduced system
\[
\dot\theta = \omega + \varepsilon Z(\theta)^\top p(t) + \cdots,\qquad
\dot r = \mu r + \varepsilon U(\theta)^\top p(t) + \cdots,
\]
where \(U(\theta)\) is the phase-dependent amplitude response function. The associated adjoint DDE for the amplitude response \(q(t)\) is
\[
\dot q(t) =
-\big({\rm D}F_0(t)^\top-\mu I_m\big)q(t)
-
e^{-\mu\tau}{\rm D}F_1(t+\tau)^\top q(t+\tau),
\qquad q(t+T)=q(t),
\]
with normalization
\[
q(0)^\top \rho(0)
+
e^{-\mu\tau}\int_{-\tau}^0 q(\tau+\zeta)^\top {\rm D}F_1(\tau+\zeta)\rho(\zeta)\,d\zeta
=1.
\]
The reduced amplitude equation makes PDA explicit: the same forcing \(p(t)\) produces different amplitude increments depending on phase through \(U(\theta)\) [2404.17356].

## 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 [2204.12783], the received signal through an RIS is
\[
y_t = \bar\alpha \sum_{m=1}^M [b(r)]_m w_{t,m} s_t + n_t,
\qquad
w_{t,m} = \beta(\theta_{t,m};\eta)e^{j\theta_{t,m}},
\]
with
\[
\beta(\theta;\eta) = (1-\beta_{\min})\left[\frac{\sin(\theta-\phi)+1}{2}\right]^\kappa + \beta_{\min}.
\]
If the receiver assumes a unit-amplitude RIS response \( \tilde w_{t,m}=e^{j\theta_{t,m}} \), model mismatch arises [2204.12783].

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 \(30\) dB SNR in the reported setup, the lower bound grows from approximately \(0.021\) m at \(\beta_{\min}=1\) to approximately \(0.064\) m at \(\beta_{\min}=0.7\) and approximately \(0.216\)–\(0.333\) m as \(\beta_{\min}\to 0\), whereas the correctly specified CRB remains approximately \(0.021\)–\(0.022\) m [2204.12783]. The paper also states that ignoring PDA can require more than \(4\times\) more elements to match the accuracy obtained with a correctly modeled RIS [2204.12783].

To mitigate this, the paper proposes joint estimation of user location and RIS amplitude model parameters, with a joint Fisher information matrix for
\[
\chi = [\Re\{\alpha\}, \Im\{\alpha\}, r^T, \beta_{\min}, \kappa, \phi]^T,
\]
and an alternating refinement algorithm that calibrates \(\beta_{\min}\), \(\kappa\), and \(\phi\) online [2204.12783]. The reported simulation results indicate fast convergence and performance close to the CRB [2204.12783].

In HAPS-RIS-assisted MIMO systems [2604.25331], PDA enters the effective channel through two aggregated quantities,
\[
\eta_{\mathrm{RIS}} = a_{\mathrm{RIS}}^H(\phi_r,\theta_r)\Phi a_{\mathrm{RIS}}(\phi_H,\theta_H),
\qquad
\xi_\beta = \sum_{n=1}^{N_{\mathrm{RIS}}} \beta_n^2.
\]
Under a central-limit approximation, the cascaded channel entries are approximately Gaussian with mean
\[
\mu_{r,t} =
\sqrt{\frac{K_H K_G}{(K_H+1)(K_G+1)}}
\,\eta_{\mathrm{RIS}}
\,[a_R(\vartheta_r)]_r [a_T(\phi_t,\theta_t)]_t^*,
\]
and variance
\[
\Sigma =
\frac{K_H + K_G + 1}{(K_H+1)(K_G+1)}
\sum_{n=1}^{N_{\mathrm{RIS}}} \beta_n^2.
\]
The post-MRC SNR is modeled as a non-central quadratic form
\[
\rho = \bar\rho\,Q,\qquad Q=\|\mu+z\|^2,
\]
with \(z\sim \mathcal{CN}(0,R_{\mathrm{eff}})\) and
\[
R_{\mathrm{eff}} = aI + b m_r m_r^H.
\]
PDA therefore affects both the non-centrality and covariance of the SNR statistic [2604.25331].

The paper derives a saddle-point-approximation framework for the SNR distribution. The cumulant generating function is
\[
K_Q(s) =
-\sum_i \ln(1-s\lambda_i)
+
\sum_i \frac{s|\tilde\mu_i|^2}{1-s\lambda_i},
\]
with corresponding SPA expressions for the PDF, CDF, and outage probability [2604.25331]. It reports that PDA reduces coherent gain and increases outage, while higher phase resolution mitigates the penalty [2604.25331].

A related beamforming problem is studied in "Discrete Beamforming Optimization for RISs with a Limited Phase Range and Amplitude Attenuation" [2507.07342], where the RIS coefficient is
\[
\rho_n(\phi_n)=a(\phi_n)e^{j\phi_n},
\]
and the practical PDA curve is
\[
\beta^r(\theta)=
(1-\beta^r_{\min})\left(\frac{\sin(\theta-\phi^r)+1}{2}\right)^{\alpha^r}
+\beta^r_{\min}.
\]
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 \(NK\) steps [2507.07342]. 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 \(K=4\) yields only marginal gains, provided the phase range \(R\) is sufficiently wide [2507.07342].

## 7. Methods, assumptions, and limitations

Across these literatures, PDA is technically tractable only under fairly specific assumptions. In oscillator phase reduction [2305.04277], the analysis assumes weak coupling \(K\ll1\), small deformation \(\delta\ll1\), smooth \(g\), persistence of a normally hyperbolic invariant \(N\)-torus, and sufficiently strong radial attraction \(m<0\) 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 \(\omega_k\) reduces tractability, and that large \(\delta\) would invalidate the small-parameter expansion [2305.04277].

In the finite-background NLS setting [1906.00959], 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 \(\phi_\tau=0\). The paper notes that more general backgrounds may require time-dependent displaced phase \(\phi(\xi,\tau)\), rendering the oscillator non-autonomous [1906.00959].

In neural phase–amplitude DCM [1909.08509], the model assumes operation near a supercritical Hopf bifurcation with circular limit cycles and specific weighting choices \(e^{-|r_i-r_j|}\) and \(\cos(\theta_i-\theta_j)\). 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 [1909.08509].

In delay equations [2404.17356], 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 [2404.17356].

In RIS applications [2204.12783, 2604.25331, 2507.07342], PDA models are hardware calibrated and effective rather than fundamental. Their practical success depends on the fidelity of the amplitude–phase curve \(\beta(\theta)\) or \(\beta_n(\varphi_n)\), 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 [2305.04277]. In dispersive wave theory, it embeds phase into the effective amplitude potential [1906.00959]. In neural state-space models and delay equations, it enters as phase-dependent amplitude coupling or response functions [1909.08509, 2404.17356]. In RIS systems, it formalizes hardware-imposed amplitude attenuation as a function of phase and thereby reshapes inference and beamforming problems [2204.12783, 2604.25331, 2507.07342]. 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.

Source: https://www.emergentmind.com/topics/phase-dependent-amplitude-pda