Synchronization of non-weakly coupled aeroelastic oscillators

Abstract

Synchronized oscillators are ubiquitous in nature and engineering. Despite several models that have been proposed to treat synchronized oscillators beyond weak coupling, the widely accepted paradigm holds that synchronization occurs due to weak interactions between oscillating objects, hence limiting the predictive power of such models to the weak coupling limit. Here, we report a theoretical modeling and experimental observation of a synchronized pair of non-weakly coupled aeroelastic oscillators. We find quantitative agreement between the experiments and our theoretical higher-order phase model of non-weak coupling. Our results establish that synchronization experiments can be accurately reproduced and interpreted by theoretical modeling of non-weakly coupled oscillators, extending the range of validity and prediction power of theoretical phase models beyond the weak coupling limit.

Introduction

From superconductivity¹ and superfluidity² at the subatomic scale to orbital resonances of planets at celestial scales³, synchronization pervades all of nature, science, and engineering. Since Huygens’ observation in 1665 of an odd kind of sympathy between a pair of pendulum clocks, researchers made immense progress in the theory, experiments, and applications of synchronization. Examples include theoretical derivation of reduced-order phase models to analyze synchronization^4,5,6,7,8, theoretical discoveries and experimental observations of exotic synchronization phenomena, such as cluster synchronization⁹, phase waves and turbulence^10,11,12, and chimera states^13,14,15,16, analysis of complex fluid flows^17,18,19,20, usage of synchronization for frequency stabilization^21,22,23,24, for frequency synthesizers^25,26,27, and for neuromorphic computing^28,29,30. In this continuous effort to explore synchronization and stretch the envelope of its applications, there have been numerous successful attempts to model the synchronization of non-weakly coupled oscillators by using advanced and sophisticated phase reduction methods^{31,32,33,34,35}. However, the widely accepted paradigm holds that synchronization occurs due to a weak coupling (or interaction) between self-sustained oscillating objects. From this assumption of weak coupling, one can readily derive the well-known Kuramoto phase model for many globally coupled oscillators⁴ or the Adler equation³⁶, which is a special case of the Kuramoto phase model for a pair of oscillators^6,22,23.

This paper investigates the mutual synchronization of a pair of aeroelastic oscillators with a non-weak coupling. We show that the noisy Adler equation of weakly coupled oscillators gives an inaccurate prediction that misses the synchronization range by almost 20%. Unlike recent studies that use sophisticated phase reduction methods^37,38,39,40, we take a simple and straightforward approach, which can be viewed as a direct modification of the classical Haken-type phase reduction method^41,42, to derive a generalized phase model that accurately predicts the experimental measurements for non-weak coupling. Our findings show that the strict requirement of weak coupling, described by the Adler equation, can be relaxed and replaced by a higher-order phase model that not only describes the synchronization of weakly coupled oscillators but also adequately applies to larger coupling strengths.

Results and discussion

Experiments

We conducted the experiments in a low-speed wind tunnel (Fig. 1) with a test section of a 0.7 × 0.7 m² rectangular cross-section area that is 1.35 m long. The speed of the streamwise airflow in the test section can be tuned to different average values between U_∞ = 2 − 25 m/s. We anchored a pair of mechanical resonators inside the test section. Each mechanical resonator is composed of a square aluminum prism [hydraulic diameter h = 0.02 m, length ℓ_p = 0.56 m, mass m = 0.247 kg] that is vertically mounted and connected to a pair of aluminum cantilevers with nominal dimensions of 1(length) × 0.04(width) × 0.005(thickness) m³, actual adjustable length (form anchor to prism) ℓ_1,2 = 0.56 − 0.62 m, a flexural rigidity EI = 16.8 kg ⋅ m³/s², and a mass per unit length ρ = 0.46 kg/m.

Fig. 1: Coupled aeroelastic oscillators.

a Each of the two identical rigid prisms (mass m, length ℓ_p, and rectangular cross-section h × h) is connected to a pair of identical elastic aluminum cantilevers with a mass density per unit length ρ, flexural rigidity EI, and variable length ℓ_1,2. b The variable length of the cantilevers ℓ_1,2, enables control of the oscillation frequency of the prisms and can be adjusted (from anchor to prism) in the range of 0.56–0.62 m. The white scale bar shows a length of 20 cm. c The upstream speed U_∞ of the air in the streamwise direction of the test section produces transverse forces on the prisms ${F}_{{q}_{1,2}}$ that induce self-sustained oscillations of the prisms, q₁(t) and q₂(t). d The prisms interact with each other via a pair of viscoelastic springs (c_cpl, k_cpl) that are attached to the upper and lower cantilevers. The coupling strength is adjusted by changing the springs' distance from the cantilevers' clamped end ℓ_s. e The coupling springs are covered by a custom-made 3D-printed smooth pipe to prevent buckling and minimize energy dissipation due to friction and anchoring losses.

The self-sustained oscillations of the mechanical resonators occur due to a transverse aerodynamic force^43,44 that serves as a feedback loop, where beyond a threshold value of airflow speed, the added linear damping from the airflow becomes negative—such that the airflow pumps energy to the resonator. Consequently, the transverse motion of the resonator increases until nonlinear effects commence and the amplitude saturates at a stable limit cycle. We use the term aeroelastic oscillator to describe the actively self-oscillating component that includes the mechanical resonator with the transverse aerodynamic force beyond the instability threshold. We note that our aeroelastic oscillator can generate multiple coexisting limit cycles at a certain range of airflow speed; however, in our study, we used a constant airflow speed of 8 m/s in which there is a single (and stable) limit cycle (Methods).

We imposed the coupling between the oscillators by a pair of springs that are attached to the lower and upper cantilevers (Fig. 1). Each spring moves inside a custom-made 3D-printed smooth pipe, which prevents spring buckling and minimizes energy dissipation due to friction and anchoring losses. We estimated the stiffness [k_cpl = 203.250 kg/s²] and damping constant [c_cpl = 0.107 kg/s] of the viscoelastic springs from ring-down measurements (Methods), which also validate that the coupling forces are linear throughout the range of amplitudes that we consider in this paper.

We recorded the transverse motions of the prisms with a high-speed camera (IDS, UI-3140CP Rev .2) equipped with an M1214-MP2 lens at a frame rate of 1.2 KHz, which is three orders of magnitude higher than the measured oscillation frequency [ ≈ 5.55 Hz]. We processed the videos using an in-house MATLAB-based image processing routine to extract the position of the prisms as functions of time (Methods), and we used the built-in Hilbert transform of MATLAB to extract the phases ϕ_1,2(t) and amplitudes a_1,2(t) of the oscillators from the time tracks of the prisms’ positions.

Theory

We derived a reduced-order model for the amplitude of each oscillator a_1,2 and their phase difference ψ = ϕ₂ − ϕ₁ (see Supplementary Note 1), which takes the form

$${\dot{a}}_{1,2}=-F({a}_{1,2})\pm \kappa {a}_{2,1}\sin \psi +{\zeta }_{1,2}(t),$$

(1)

$$\dot{\psi }=\Delta \omega +G({a}_{1},{a}_{2})-\kappa \left(\frac{{a}_{1}}{{a}_{2}}-\frac{{a}_{2}}{{a}_{1}}\right)\cos \psi +{\zeta }_{3}(t),$$

(2)

where $F({a}_{1,2})={\sum }_{n}\,{f}_{n}{a}_{1,2}^{n}$ models the amplitude dynamics of the uncoupled aeroelastic oscillator and the saturation at a stable limit-cycle a_1,2 = a_ss, which satisfy the conditions F(a_ss) = 0 and ${F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}}) > 0$, κ is the reactive coupling strength (the dissipative coupling c_cpl is found to be negligible, see Methods), ζ_j(t) are delta-correlated Gaussian noises $\langle {\zeta }_{i}(t){\zeta }_{j}({t}^{{\prime} })\rangle =2{\sigma }_{i}^{2}{\delta }_{ij}\delta (t-{t}^{{\prime} })$ of low intensity $({\sigma }_{i}^{2})$ that account for the turbulence-induced random fluctuations, Δω = ω₂ − ω₁ is the frequency mismatch of the uncoupled aeroelastic oscillators, and $G({a}_{1},{a}_{2})={\sum }_{n}{g}_{n}({a}_{2}^{2n}-{a}_{1}^{2n})$ encapsulates the difference in the frequency pulling of the uncoupled aeroelastic oscillators (see Supplementary Note 1).

To derive a phase model for ψ, we take a perturbative approach, where we set ${a}_{1,2}={a}_{{{{{{{{\rm{ss}}}}}}}}}+{\sum }_{n}{\kappa }^{n}\delta {a}_{1,2}^{(n)}$, assume that amplitude fluctuations are very small ${\zeta }_{1,2} \sim O({\kappa }^{3})$, and find the following expansions for the steady-state amplitudes (see Methods)

$${a}_{1,2}={a}_{{{{{{{{\rm{ss}}}}}}}}}\pm \frac{\kappa {a}_{{{{{{{{\rm{ss}}}}}}}}}}{{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})}\sin \psi -\frac{{\kappa }^{2}{a}_{{{{{{{{\rm{ss}}}}}}}}}[2{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})+{a}_{{{{{{{{\rm{ss}}}}}}}}}{F}^{{\prime\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})]}{2{F}^{{\prime} 3}({a}_{{{{{{{{\rm{ss}}}}}}}}})}{\sin }^{2}\psi +O({\kappa }^{3}),$$

(3)

where ${F}^{{\prime} }$ and F^″ are the first and second derivatives of F with respect to the amplitude. Next, we substituted the expansions of a_1,2 into Eq. (2), and find that, to order $O({\kappa }^{2})$, the phase model for ψ is given by

$$\dot{\psi }=\Delta \omega -{c}_{1}\kappa \sin \psi -{c}_{2}{\kappa }^{2}\sin 2\psi +{\zeta }_{3}(t),$$

(4)

where ${c}_{1}=[{a}_{{{{{{{{\rm{ss}}}}}}}}}/{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})]({\partial }_{{a}_{2}}G-{\partial }_{{a}_{1}}G){| }_{{a}_{{{{{{{{\rm{ss}}}}}}}}}}$, and ${c}_{2}=[2/{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})]$.

Before we proceed to the Eq. (4) analysis, we make some remarks on the assumptions and validity of Eqs. (3)–(4). Strictly speaking, Eqs. (3)–(4) are valid only inside the synchronization range, where $\Delta \omega \sim O(\kappa )$, ψ(t) is a slowly varying function of time [$\dot{\psi } \sim O(\kappa )$], and a_1,2(t) decay rapidly [${\dot{a}}_{1,2} \sim O({F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}}))$] towards their pseudo steady-state values in Eq. (3) for small deviations from a_ss. Under these conditions, the dynamical system in Eqs. (1)–(2) exhibit slow-fast dynamics in which after a short time [of order $1/{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})$], i.e., the fast part of the dynamics, the effects of initial conditions of the amplitudes a_1,2(0) are attenuated to zero, and the amplitudes track the phase difference adiabatically as described in Eq. (3), which is the slow part of the dynamics. It is important to note that Eqs. (3)–(4) represent the dynamics of Eqs. (1)–(2) accurately only in the slow part of the dynamics, where $t\gg 1/{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})$, and under the condition that there are two distinct time scales, ${T}_{{{{{{{{\rm{slow}}}}}}}}} \sim O(1/\kappa )$ and ${T}_{{{{{{{{\rm{fast}}}}}}}}} \sim O(1/{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}}))$, where ${F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})\gg \kappa$. While the above assumptions limit the range of Δω, the phase model [Eq. (4)] is also valid when ∣Δω∣ ≫ κ and the phase difference runs freely $\dot{\psi }\approx \Delta \omega$, i.e., when the oscillators are effectively uncoupled.

Analysis

Inspection of Eq. (4) reveals that the phase difference dynamics can be mapped onto the noisy motion of an overdamped particle in a tilted washboard potential $V(\psi )=-\Delta \omega \psi -{c}_{1}\kappa \cos \psi -({c}_{2}{\kappa }^{2}/2)\cos 2\psi$, see Methods. Furthermore, for small κ (weak coupling), Eq. (4) reduces to the well-known Adler equation^6,22,23. However, for non-small κ (non-weak coupling), the additional term $-{c}_{2}{\kappa }^{2}\sin 2\psi$, can dramatically change the synchronization dynamics, including the birth of an additional pair of synchronized states (i.e., four extrema in $V$ in each 2π interval) and enhanced synchronization range relative to the range of the Adler equation (Methods). We note that under the experimental conditions that we considered [U_∞ = 8 m/s, and ℓ_s = 0.4 m], where κ = 0.314, c₁ = 0.036, and c₂ = 0.043, we only obtained an enhanced synchronization range (Supplementary Fig. 1). Moreover, by carrying the perturbation analysis to cubic order in κ, we find that the cubic order correction contains the additional terms $-{c}_{3}{\kappa }^{3}{\sin }^{3}\psi +{r}_{1}{\zeta }_{1}(t)+{r}_{2}{\zeta }_{2}(t)$; see Methods for details. Thus, the higher-order phase model in Eq. (4) is valid only when c₂ ≫ κc₃ and ∣ζ₃∣ ≫ ∣r_1,2ζ_1,2∣.

The current of the phase difference $\langle \dot{\psi }\rangle ={\lim }_{t\to \infty }\langle \psi \rangle /t$ can be computed from the following expression⁴⁵

$$\langle \dot{\psi }\rangle =\frac{2\pi {\sigma }_{3}^{2}\left(1-{{{{{{{{\rm{e}}}}}}}}}^{-2\pi \Delta \omega /{\sigma }_{3}^{2}}\right)}{\int_{0}^{2\pi }\int_{\psi -2\pi }^{\psi }{{{{{{{{\rm{e}}}}}}}}}^{-[V(\varphi )-V(\psi )]/{\sigma }_{3}^{2}}d\varphi d\psi },$$

(5)

which we estimate numerically (Supplementary Note 2). In Fig. 2, we show the theoretical predictions of the normalized averaged amplitudes 〈a_1,2〉/a_ss and the current $\langle \dot{\psi }\rangle$ from Eq. (3) and Eq. (5), along with the predictions from the Adler equation⁴⁶, and experimental data. The predictions of the Adler equation fail to describe the amplitude dependency on the frequency mismatch and the synchronization range (the predicted range is almost 20% shorter), while the higher-order model of Eq. (4) gives accurate predictions for the experiments.

Fig. 2: Synchronization of aeroelastic oscillators with a non-weak coupling.

Comparison between experimental data (points) and theoretical predictions from the higher-order model [Eqs. (3)–(4), black curve] and the Adler equation [Eqs. (3)–(4) truncated at order $O(\kappa )$, gray curve]. The theoretical predictions for the amplitudes (a) exist only in the synchronized range (shaded region), where the current of the phase difference (b) is zero. The vertical dashed, dotted, and dashed-dotted lines correspond to the experimental data shown in the top, center, and bottom panels of Fig. 3, respectively.

To analyze the motion of the oscillators and distinguish between synchronized state (zero current $\langle \dot{\psi }\rangle \approx 0$), free-running state (non-zero current $\langle \dot{\psi }\rangle \ne 0$), and injection pulling, which is a transition state between free-running and synchronized states, we represent the measured signals in several domains. In Fig. 3, we show three samples from the experimental data of these three distinct states. The representations of the oscillators’ signals in the time (panels a–c) and frequency (panels d–f) domains show distinctively different patterns that are readily identified with the different states of the oscillators. Furthermore, the construction of a torus from the total measured phases of the oscillators and a Poincaré map (panels g-i) reveals that unlike the dense toroidal phase space and Poincaré map of the free-running oscillators and injection pulling, in the synchronized state, the phase space and Poincaré map are sparse (signal trajectory and single fixed point).

Fig. 3: Analysis of the oscillators’ motions.

The experimentally measured displacements of the oscillators are shown in the time domain (a–c), frequency domain (d–f), toroidal phase space θ_1,2 = ω_1,2t + ϕ_1,2 and its respective Poincaré map θ_1p(k) = θ₁(t_k),  θ_2p(k) = θ₂(t_k) = 2πk (g–i). For large frequency mismatch (dashed line in Fig. 2), the coupling effect is negligible, and oscillators run freely (a, d, g) with a small mutual injection of motion that can only be detected in the power spectra on dB scales. For moderate frequency mismatch (dotted line in Fig. 2), the coupling-induced injected motion pulls the oscillators (b, e, h) toward synchronization. This injection pulling is highly visible both in time (beating envelopes) and frequency (side-bands with a spacing of the beats frequency) domains. For small frequency mismatch (dashed-dotted line in Fig. 2), the oscillators synchronized (c, f, i). The phase-locked synchronized motion is readily identified in the time and frequency domains, and associated with a unique trajectory in the toroidal phase space and a single fixed point in the Poincaré map.

Conclusions

Our demonstration of synchronizing two non-weakly coupled aeroelastic oscillators shows excellent agreement with analytical and numerical predictions. We experimentally control not only the frequency mismatch but also the coupling strength, which is essential for a complete description of the non-weakly coupled synchronized oscillators. The relatively simple macroscale mechanical system we used is easily characterized and tuned to operate at the desired conditions of a weakly nonlinear dynamical system with non-weak coupling (see Methods). Our results highlight the importance of non-weak coupling for enhanced synchronization range, which can be highly beneficial for various applications ranging from wireless communications⁴⁷ and time-keeping²⁴ to quantum networks⁴⁸. Our results extend the synchronization problem beyond weak coupling, relaxing the strict weak coupling requirement. This is achieved by replacing the Adler equation with a higher-order phase model that successfully describes the synchronization of two oscillators across a broad range of coupling strengths.

Methods

Measurements and characterization

To measure the transverse motions of the prisms, we marked each prism with duct tape, which appeared as a distinct rectangle region in the images, and we tracked its centroid to obtain the position of the prism as a function of time. We converted the pixel measurements to meters using a conversion meters-per-pixel (MPP) equation, which accurately accounts for the change in the length of the cantilevers. We calculated the MPP ratio by measuring the pixels at different lengths and utilizing the known width of the prisms. We then applied a trend line, resulting in the equation that converts the number of pixels to the displacement of each prism (see Supplementary Fig. 2). This process enabled a clear representation of the motions of the prisms, which was subsequently ready for analysis.

We estimated the linear, quadratic, and cubic damping coefficients in Supplementary Table 1 by analyzing the mechanical resonators’ ring-down responses^49,50. We partitioned the decaying envelope of the signal into three amplitude regions (low, moderate, and high) and used the same technique of ref. ⁴⁴ to estimate Γ_1,2,3. Using the estimated damping coefficient Γ_1,2,3, the calculated resonators’ properties ω_1,2, m_r, η (Supplementary Table 1), and coefficients of the aerodynamic force ν_n, μ_n (based on the Parkinson model⁵¹, Supplementary Table 2, we constructed a theoretical amplitude response curve. We validated it with experimental measurements for several cantilever lengths (Fig. 4a). These validated response curves include multiple coexisting limit cycles (which we measured by sweeping the wind speed up and down) and variability of the limit-cycle amplitude as a function of the cantilevers’ length. Therefore, we pick a wind speed of 8 m/s as the desired operating point at which the amplitudes of the various oscillators are relatively close to one another (less than 8% difference), and there are no multiple coexisting limit cycles. At this wind speed, we measure the steady-state power spectrum of the oscillator around its carrier frequency f − f_osc and use a Lorentzian fit to extract the phase diffusion D (Fig. 4b). We note that the Lorentzian distribution only describes pure phase diffusion (i.e., negligible amplitude fluctuations)⁵², and the collapse of all experimental curves on the same Lorentzian distribution indicates a constant intensity of turbulence-induced random fluctuations (or, alternatively, that the delta-correlated Gaussian noise representation of the turbulence-induced random fluctuations in the amplitudes and phase equations is appropriate). Moreover, the noise intensity in Eq. (2) is proportional to the measured phase diffusion ${\sigma }_{3}^{2}=2D/{\omega }_{s}=5.23\times 1{0}^{-3}$.

Fig. 4: Steady-state response of a single aeroelastic oscillator.

Experimental measurements of a single aeroelastic oscillator with various cantilever lengths (measured in meters): 0.62 (black), 0.59 (light gray), 0.56 (dark gray). The amplitude response curves exhibit a region of multiple coexisting limit cycles (stable/unstable denoted by solid/dashed curve) and non-identical responses for different cantilever lengths (a). The spectrum of the aeroelastic oscillator around its carrier frequency is almost identical regardless of the cantilevers' lengths (b) and closely follows the Lorentzian distribution (dashed blue curve).

For the coupling coefficients estimations (Supplementary Table 3), we perturbed each mechanical resonator from its equilibrium position in still air while holding the second resonator in its equilibrium position. By comparing the free oscillations decay toward the equilibrium position with and without coupling (Fig. 5a, b), we were able to extract the coupling-induced stiffness and damping. In particular, from the frequency shift Δf = f_coupled − f_uncoupled, we extracted the coupling stiffness Δf = ω_sκ/(2π), and from the slope of the decaying amplitude (at the linear decay regime), we extracted the coupling damping coefficient $\chi =d\ln [{a}_{{{{{{{{\rm{coupled}}}}}}}}}(t)/{a}_{{{{{{{{\rm{uncoupled}}}}}}}}}(t)]/dt$, where χ = ϕ²(ℓ_s/ℓ)c_cpl/[ρℓ(1 + 2μ)]. We note that Δf remained approximately constant (Fig. 5c) during the ring-down experiments for the various set-ups we considered, and hence, we deduce that the reactive coupling is linear. The difference in the decaying amplitude in the presence and absence of coupling is relatively small (Fig. 5d), and we could only measure a slight difference in the linear decay, which revealed that κ ≫ χ/ω_s, and hence, that the dissipative coupling is negligible with respect to the reactive coupling.

Fig. 5: The ring down response.

Estimation of coupling from the ring down response for ℓ = 0.59 m and ℓ_s = 0.25 m. By measuring the ring-down response (a) with/without (gray/black signals) coupling, we detect the zero-crossing of the signals (b) and estimate the frequency (c) and amplitude (d) as functions of time. We estimate the reactive and dissipative coupling coefficients from the difference in frequency and amplitude.

Higher-order phase model

As described in subsection II B, we use a standard perturbation method to obtain the higher-order phase model. That is, we calculate the small perturbations of the expansion ${a}_{1,2}={a}_{{{{{{{{\rm{ss}}}}}}}}}+{\sum }_{n}{\kappa }^{n}\delta {a}_{1,2}^{(n)}$ by substituting the expansion into Eq. (1) and equating terms of equal powers of κ to get the following set of equations

$$O(1):F({a}_{{{{{{{{\rm{ss}}}}}}}}})=0,$$

(6)

$$O(\kappa ):\delta {\dot{a}}_{1,2}^{(1)}=-{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})\delta {a}_{1,2}^{(1)}\pm {a}_{{{{{{{{\rm{ss}}}}}}}}}\sin \psi ,$$

(7)

$$O({\kappa }^{2}):\,\delta {\dot{a}}_{1,2}^{(2)}=-{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})\delta {a}_{1,2}^{(2)}\pm \delta {a}_{2,1}^{(1)}\sin \psi -\frac{1}{2}{\left(\delta {a}_{1,2}^{(1)}\right)}^{2}{F}^{{\prime\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}}),$$

(8)

$$\begin{array}{rcl}&&O({\kappa }^{3}):\,\delta {\dot{a}}_{1,2}^{(3)}=-{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})\delta {a}_{1,2}^{(3)}\pm \delta {a}_{2,1}^{(2)}\sin \psi \\ &&-\delta {a}_{1,2}^{(1)}{a}_{1,2}^{(2)}{F}^{{\prime\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})-\frac{1}{6}{\left(\delta {a}_{1,2}^{(1)}\right)}^{3}{F}^{{\prime\prime\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})+{\zeta }_{1,2}^{({{{{{{{\rm{rs}}}}}}}})}(t),\end{array}$$

(9)

where ${\zeta }_{1,2}(t)={\kappa }^{3}{\zeta }_{1,2}^{{{{{{{{\rm{(rs)}}}}}}}}}(t)$. The zero-order equation, Eq. (6), is the steady-state equation of the uncoupled oscillators, which is satisfied immediately. We see that the perturbations $\delta {a}_{1,2}^{(n)}$ of the steady-state amplitude a_ss are strongly damped [${F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})$ is the effective damping constant], and so, for $t\, \gg \, 1/{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})$, we can assume $\delta {\dot{a}}_{1,2}^{(n)}\approx 0$ (see Appendix B of ref. ⁵³ for detailed proof). Thus, from the first-order equation, Eq. (7), we find that $\delta {a}_{1,2}^{(1)}=\pm {a}_{{{{{{{{\rm{ss}}}}}}}}}\sin \psi /{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})$, which we substitute into Eq. (8) to obtain $\delta {a}_{1,2}^{(2)}=-{a}_{{{{{{{{\rm{ss}}}}}}}}}[2{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})+{a}_{{{{{{{{\rm{ss}}}}}}}}}{F}^{{\prime\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})]{\sin }^{2}\psi /[2{F}^{{\prime} 3}({a}_{{{{{{{{\rm{ss}}}}}}}}})].$ Finally, we substitute $\delta {a}_{1,2}^{(1)}$ and $\delta {a}_{1,2}^{(2)}$ into Eq. (9) to find that $\delta {a}_{1,2}^{(3)}={\zeta }_{1,2}^{({{{{{{{\rm{rs}}}}}}}})}(t)/{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})\mp {a}_{{{{{{{{\rm{ss}}}}}}}}}\{6{F}^{{\prime} 2}({a}_{{{{{{{{\rm{ss}}}}}}}}})-3{a}_{{{{{{{{\rm{ss}}}}}}}}}^{2}{F}^{{\prime\prime} 2}({a}_{{{{{{{{\rm{ss}}}}}}}}})+{a}_{{{{{{{{\rm{ss}}}}}}}}}{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})[{a}_{{{{{{{{\rm{ss}}}}}}}}}{F}^{{\prime\prime\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})-3{F}^{{\prime\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})]\}{\sin }^{3}\psi /[6{F}^{{\prime} 5}({a}_{{{{{{{{\rm{ss}}}}}}}}})].$By substituting the expansion of the amplitudes into Eq. (2), we find that

$$\dot{\psi }= \, \Delta \omega -{c}_{1}\kappa \sin \psi -{c}_{2}{\kappa }^{2}\sin 2\psi -{c}_{3}{\kappa }^{3}{\sin }^{3}\psi \\ +{r}_{1}{\zeta }_{1}(t)+{r}_{2}{\zeta }_{2}(t)+{\zeta }_{3}(t),$$

(10)

where

$${r}_{1,2} = \frac{({\partial }_{{a}_{1,2}}G){| }_{{a}_{{{{{{{{\rm{ss}}}}}}}}}}}{{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})},\,{c}_{1}=\frac{{a}_{{{{{{{{\rm{ss}}}}}}}}}}{{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})}({\partial }_{{a}_{2}}G-{\partial }_{{a}_{1}}G){| }_{{a}_{{{{{{{{\rm{ss}}}}}}}}}},\\ {c}_{2} =\frac{2}{{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})},\,{c}_{3}=\frac{{a}_{{{{{{{{\rm{ss}}}}}}}}}}{6{F}^{{\prime} 5}({a}_{{{{{{{{\rm{ss}}}}}}}}})}\left\{3{a}_{{{{{{{{\rm{ss}}}}}}}}}^{2}{F}^{{\prime\prime} 2}({a}_{{{{{{{{\rm{ss}}}}}}}}}){[{\partial }_{{a}_{2}}G-{\partial }_{{a}_{1}}G]}_{{a}_{{{{{{{{\rm{ss}}}}}}}}}}\right.\\ \quad+{a}_{{{{{{{{\rm{ss}}}}}}}}}{F}^{{\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})\left[{a}_{{{{{{{{\rm{ss}}}}}}}}}{F}^{{\prime\prime\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})({\partial }_{{a}_{1}}G-{\partial }_{{a}_{2}}G){| }_{{a}_{{{{{{{{\rm{ss}}}}}}}}}}\right.\\ \left.\quad+\left.3{F}^{{\prime\prime} }({a}_{{{{{{{{\rm{ss}}}}}}}}})\left({\partial }_{{a}_{2}}G-{\partial }_{{a}_{1}}G-{a}_{{{{{{{{\rm{ss}}}}}}}}}{\partial }_{{a}_{2}}^{2}G+{a}_{{{{{{{{\rm{ss}}}}}}}}}{\partial }_{{a}_{1}}^{2}G\right)\right| _{{a}_{{{{{{{{\rm{ss}}}}}}}}}}\right]\\ \quad+{F}^{{\prime} 2}({a}_{{{{{{{{\rm{ss}}}}}}}}})\left[6({\partial }_{{a}_{1}}G-{\partial }_{{a}_{2}}G){| }_{{a}_{{{{{{{{\rm{ss}}}}}}}}}}\right.\\ \quad \left. +\left.\left.{a}_{{{{{{{{\rm{ss}}}}}}}}}\left(6{\partial }_{{a}_{1}}^{2}G-6{\partial }_{{a}_{2}}^{2}G-{a}_{{{{{{{{\rm{ss}}}}}}}}}{\partial }_{{a}_{1}}^{3}G+{a}_{{{{{{{{\rm{ss}}}}}}}}}{\partial }_{{a}_{2}}^{3}G\right)\right| _{{a}_{{{{{{{{\rm{ss}}}}}}}}}}\right]\right\}.$$

Thus, when c₂ ≫ κc₃ and ∣ζ₃∣ ≫ ∣r_1,2ζ_1,2∣, Eq (10) reduces to Eq. (4), i.e., to the higher-order phase model that we analyze in this paper.

Eq. (4) can be written compactly as

$$\dot{\psi }=-\frac{dV}{d\psi }+{\zeta }_{3}(t),$$

(11)

where

$$V(\psi )=-\Delta \omega \psi -{c}_{1}\kappa \cos \psi -\frac{{c}_{2}{\kappa }^{2}}{2}\cos 2\psi$$

(12)

is an effective washboard potential, and its local minima correspond to stable synchronized states. From Eq. (12), we find that if the condition $d=16{c}_{2}^{2}{\kappa }^{2}-{c}_{1}(\sqrt{32{c}_{2}^{2}{\kappa }^{2}+{c}_{1}^{2}}+{c}_{1}) \, > \, 0$ is satisfied, then the potential $V(\psi )$ has in each 2π interval: two pairs of stable and unstable synchronized states when $| \Delta \omega | < | \Delta {\omega }_{{{{{{{{\rm{cr}}}}}}}}}^{(1)}|$ (Fig. 6a), a single pair of stable and unstable synchronized states when $| \Delta {\omega }_{{{{{{{{\rm{cr}}}}}}}}}^{(1)}| < | \Delta \omega | < | \Delta {\omega }_{{{{{{{{\rm{cr}}}}}}}}}^{(2)}|$ (Fig. 6b), and no synchronized states when $| \Delta {\omega }_{{{{{{{{\rm{cr}}}}}}}}}^{(2)}| < | \Delta \omega |$ (Fig. 6c). The critical values of the frequency mismatch $\Delta {\omega }_{{{{{{{{\rm{cr}}}}}}}}}^{(1,2)}$ in which saddle-node bifurcations occur, are given by

$$\Delta {\omega }_{{{{{{{{\rm{cr}}}}}}}}}^{(1,2)} < \frac{\left(\sqrt{32{c}_{2}^{2}{\kappa }^{2}+{c}_{1}^{2}}\mp 3{c}_{1}\right)}{16\sqrt{2}{c}_{2}} \times \sqrt{16{c}_{2}^{2}{\kappa }^{2}\mp {c}_{1}\left(\sqrt{32{c}_{2}^{2}{\kappa }^{2}+{c}_{1}^{2}}\pm {c}_{1}\right)}.$$

(13)

If the condition d > 0 is not satisfied (as in our experiments, where d < 0), then we can have, at most (when $| \Delta \omega | < | \Delta {\omega }_{{{{{{{{\rm{cr}}}}}}}}}^{(2)}|$), a single pair of stable and unstable synchronized states.

Fig. 6: Extrema of the washboard potential.

The local extrema of $V(\psi )$ [Eq. (12)] in each 2π interval corresponds to synchronized states, where local minima are stable states, and local maxima are unstable states. If d > 0, then for small values of Δω, there are two pairs of stable and unstable synchronized states (a); for moderate values of Δω, there is a single pair of stable and unstable synchronized states (b), and there are no synchronized states for large values of Δω (c).