1. Introduction
Limit cycle oscillations are observed across a wide range of phenomena, including chemical reactions, human behaviour, circadian rhythms, and fluid flows. To shed light on the mechanism of phase synchronisation about the limit cycles, there have been extensive research efforts in physics and biology (Kuramoto Reference Kuramoto1984; Pikovsky, Rosenblum & Kurths Reference Pikovsky, Rosenblum and Kurths2001; Ermentrout & Terman Reference Ermentrout and Terman2010). The examination of phase synchronisation can be facilitated by the phase reduction analysis (Nakao Reference Nakao2016), which is a theoretical approach that reduces a high-dimensional dynamical system about a limit cycle to a single scalar phase variable (or a small number of variables) to describe the oscillatory dynamics. This technique is immensely powerful in explaining synchronisation dynamics of various complex systems.
Let us consider a dynamical system for a state variable
$\boldsymbol{q}(\boldsymbol{x},t)$
that satisfies
If this system admits a time-periodic solution
$\boldsymbol{q}_0(\boldsymbol{x},t) = \boldsymbol{q}_0(\boldsymbol{x},t+T)$
with period
$T$
, the associated phase variable
$\theta (t)$
for the limit cycle can be established. This phase variable can be defined in an appropriately chosen phase space. We can extend
$\theta (t)$
to be a function of the state variable
$\boldsymbol{q}$
in the basin of attraction of the limit cycle such that
$\theta = \varTheta (\boldsymbol{q}(\boldsymbol{x},t))$
. In this case, the phase function satisfies
where
$\omega _n = 2\pi /T$
is the natural frequency of the limit cycle oscillation.
The influence of small external forcing
$\epsilon \boldsymbol{p}(\boldsymbol{x},t)$
on the phase variable can be examined for
where
$\epsilon \ll 1$
. Under this forcing, the phase variable evolves as
In (1.4), we linearised the dynamics about the limit cycle and defined
$\boldsymbol{Z}(\theta ) \equiv \boldsymbol{\nabla} _{\boldsymbol{q}_0} \varTheta$
, known as the phase sensitivity function (PSF). This PSF can be determined computationally or experimentally from an impulse response, or computationally through adjoint analysis.
Note that (1.2) and (1.4) reduce a high-dimensional system with variable
$\boldsymbol{q}$
to its phase dynamics described by a single variable
$\theta$
, effectively capturing the oscillatory dynamics. This PSF provides significant insights into the dynamics. The PSF reveals the sensitivity of the phase variable in response to a perturbation over time and space, and allows us to find the phase synchronisation condition, which is widely known as the Arnold tongue on the frequency-amplitude map. Moreover, the phase-based model can be used to design control schemes to alter the oscillatory dynamics of the system (Monga et al. Reference Monga, Wilson, Matchen and Moehlis2019), as hinted by the inner product between the PSF and the external input
$\boldsymbol{p}$
in altering
$\dot \theta$
in (1.4). There are also networked oscillator-based models that combine multiple limit cycle representations.
2. Phase-reduction analysis of fluid flows
For fluid flows, oscillatory behaviour is prevalent in many forms, including vortex shedding, buffeting, fluttering, and flow-induced vibrations. The governing equation (1.1) for fluids can be the Navier–Stokes equations or any other equations that describe the flow physics of interest. In vortical flows, the PSF reveals the timing for the system to be receptive to phase advancement or delay, which can be directly related to the acceleration or delay of vortex formation, respectively. Phase reduction analysis has been used to examine phase synchronisation between flow unsteadiness and unsteady heat injection (Kawamura & Nakao Reference Kawamura and Nakao2015), flow control inputs (Taira & Nakao Reference Taira and Nakao2018; Nair et al. Reference Nair, Taira, Brunton and Brunton2021), and body motion (Loe et al. Reference Loe, Nakao, Jimbo and Kotani2021; Khodkar, Klamo & Taira Reference Khodkar, Klamo and Taira2021; Kim et al. Reference Kim, Godavarthi, Rolandi, Klamo and Taira2024). While the phase variable can be defined in many ways that are suitable for analysing the given problem, the force coefficient and its time derivative have been widely used.
Figure 1. (a) Flutter problem studied by Sumanasiri, Sahu & Nair (Reference Sumanasiri, Sahu and Nair2025). (b) Phase variable
$\theta (t)$
defined by the pitch angle and its time derivative with the representative flow fields shown. Figures courtesy of the authors.
Phase-reduction analysis is not limited to single-component systems. It can also be applied to multi-physics problems. In the recent paper by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025), the fluid–structure interaction problem of flutter in low-Reynolds-number flow is computationally analysed. They study the occurrence of flutter for a lightweight NACA 0015 aerofoil that is free to pitch about its one-third chord location at a chord-based Reynolds number of 1000, as illustrated in figure 1. By characterising the phase response function (reported as the PSF in their figures 3c and 3d) through impulsive modification of the spring stiffness, they find that the phase dynamics is most receptive to changes between the maximum pitching amplitude and the return to the equilibrium position. They also observe interesting departures of the flutter dynamics from the basin of attraction, as evident from the peaks appearing in the phase response functions. In their work, they make extensive connections between the insights gained from phase reduction and the flow physics by incorporating modal analysis and vortical force analysis.
To prevent the aerofoil pitching dynamics from departing from the basin of attraction, Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) further considered introducing heaving. They quantified the PSF for heaving motion and determined a phase-based control strategy that disrupts the coherence between the structural motion and unsteady aerodynamic forcing. They succeeded with this approach and were able to suppress the violent flutter. This overall suppression of flutter was achieved not only by directly applying linear phase-reduction analysis but also by cleverly leveraging physical insights gained from analysing the nonlinear vortex dynamics around the aerofoil. They have also incorporated energy transfer assessment between the flow and structural dynamics to extend their findings.
Moreover, the work by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) is refreshing in light of the growing interest in using novel structural properties to modify the dynamics of fluid flows around bodies. The impulsive modification of material properties requires additional examination to theoretically quantify its parametric effects on the phase dynamics. Such an examination invites new research initiatives to further advance fluid–structure interaction research. In fact, metamaterials may play a role in fluid–structure interaction in a passive or active manner (Bertoldi et al. Reference Bertoldi, Vitelli, Christensen and van Hecke2017; Coulais, Sounas & Alù Reference Coulais, Sounas and Alù2017). By understanding the sensitivity of the phase dynamics at precise instances in relationship to dynamical events, novel metamaterial-based control strategies as well as structural designs may be developed. The formulation and examination process presented by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) will likely stimulate these new research activities.
3. Summary and outlook
The recent work by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) demonstrated the use of phase-reduction analysis on a fluid–structure interaction problem involving flutter. By combining the insights from phase-reduction analysis with modal analysis, the authors offered deep insights into the mechanism of synchronisation (desynchronisation) and their use in suppressing undesirable flutter oscillations. Their study encourages future applications of similar approaches to complex multi-physics problems in which oscillatory dynamics play a key role.
Compression of high-dimensional dynamical systems to their phase dynamics can also be systematically aided by the use of machine-learning-based techniques. Recently, nonlinear autoencoders have been used to uncover latent-variable dynamics around limit cycles (Fukami, Nakao & Taira Reference Fukami, Nakao and Taira2024; Yawata et al. Reference Yawata, Sakuma, Fukami, Taira and Nakao2025). The discovery of low-dimensional representations through modern data-driven techniques can facilitate the phase-based modelling and control efforts not only for oscillatory flows, but also for transient flows (Fukami et al. Reference Fukami, Nakao and Taira2024). These methods can serve as a foundation to help develop phase-based models for complex multi-frequency turbulent flows.
There are open questions regarding how phase reduction analysis can be applied to turbulent flows with dominant oscillation frequencies. Since such flows do not possess an exact limit cycle, the application of phase reduction analysis requires extensions to quantify the uncertainty in phase and amplitude around the notional cyclic dynamics. This may be achieved by combining phase reduction analysis with modal analysis, such as the dynamic mode decomposition or resolvent analysis, as considered by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025). Furthermore, establishing systematic approaches to analyse energy transfers between oscillatory modes or subsystems can serve as an important tool for studying complex multi-component systems. Developments in phase-based analysis have the potential to offer novel reduced-order models and analysis techniques that can not only provide insights into unsteady physical mechanisms, but also serve as a foundation for aerodynamic and structural design optimisation, as the work by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) foreshadows.
$\theta (t)$ defined by the pitch angle and its time derivative with the representative flow fields shown. Figures courtesy of the authors.
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1. Introduction
Limit cycle oscillations are observed across a wide range of phenomena, including chemical reactions, human behaviour, circadian rhythms, and fluid flows. To shed light on the mechanism of phase synchronisation about the limit cycles, there have been extensive research efforts in physics and biology (Kuramoto Reference Kuramoto1984; Pikovsky, Rosenblum & Kurths Reference Pikovsky, Rosenblum and Kurths2001; Ermentrout & Terman Reference Ermentrout and Terman2010). The examination of phase synchronisation can be facilitated by the phase reduction analysis (Nakao Reference Nakao2016), which is a theoretical approach that reduces a high-dimensional dynamical system about a limit cycle to a single scalar phase variable (or a small number of variables) to describe the oscillatory dynamics. This technique is immensely powerful in explaining synchronisation dynamics of various complex systems.
Let us consider a dynamical system for a state variable
$\boldsymbol{q}(\boldsymbol{x},t)$
that satisfies
If this system admits a time-periodic solution
$\boldsymbol{q}_0(\boldsymbol{x},t) = \boldsymbol{q}_0(\boldsymbol{x},t+T)$
with period
$T$
, the associated phase variable
$\theta (t)$
for the limit cycle can be established. This phase variable can be defined in an appropriately chosen phase space. We can extend
$\theta (t)$
to be a function of the state variable
$\boldsymbol{q}$
in the basin of attraction of the limit cycle such that
$\theta = \varTheta (\boldsymbol{q}(\boldsymbol{x},t))$
. In this case, the phase function satisfies
where
$\omega _n = 2\pi /T$
is the natural frequency of the limit cycle oscillation.
The influence of small external forcing
$\epsilon \boldsymbol{p}(\boldsymbol{x},t)$
on the phase variable can be examined for
where
$\epsilon \ll 1$
. Under this forcing, the phase variable evolves as
In (1.4), we linearised the dynamics about the limit cycle and defined
$\boldsymbol{Z}(\theta ) \equiv \boldsymbol{\nabla} _{\boldsymbol{q}_0} \varTheta$
, known as the phase sensitivity function (PSF). This PSF can be determined computationally or experimentally from an impulse response, or computationally through adjoint analysis.
Note that (1.2) and (1.4) reduce a high-dimensional system with variable
$\boldsymbol{q}$
to its phase dynamics described by a single variable
$\theta$
, effectively capturing the oscillatory dynamics. This PSF provides significant insights into the dynamics. The PSF reveals the sensitivity of the phase variable in response to a perturbation over time and space, and allows us to find the phase synchronisation condition, which is widely known as the Arnold tongue on the frequency-amplitude map. Moreover, the phase-based model can be used to design control schemes to alter the oscillatory dynamics of the system (Monga et al. Reference Monga, Wilson, Matchen and Moehlis2019), as hinted by the inner product between the PSF and the external input
$\boldsymbol{p}$
in altering
$\dot \theta$
in (1.4). There are also networked oscillator-based models that combine multiple limit cycle representations.
2. Phase-reduction analysis of fluid flows
For fluid flows, oscillatory behaviour is prevalent in many forms, including vortex shedding, buffeting, fluttering, and flow-induced vibrations. The governing equation (1.1) for fluids can be the Navier–Stokes equations or any other equations that describe the flow physics of interest. In vortical flows, the PSF reveals the timing for the system to be receptive to phase advancement or delay, which can be directly related to the acceleration or delay of vortex formation, respectively. Phase reduction analysis has been used to examine phase synchronisation between flow unsteadiness and unsteady heat injection (Kawamura & Nakao Reference Kawamura and Nakao2015), flow control inputs (Taira & Nakao Reference Taira and Nakao2018; Nair et al. Reference Nair, Taira, Brunton and Brunton2021), and body motion (Loe et al. Reference Loe, Nakao, Jimbo and Kotani2021; Khodkar, Klamo & Taira Reference Khodkar, Klamo and Taira2021; Kim et al. Reference Kim, Godavarthi, Rolandi, Klamo and Taira2024). While the phase variable can be defined in many ways that are suitable for analysing the given problem, the force coefficient and its time derivative have been widely used.
Figure 1. (a) Flutter problem studied by Sumanasiri, Sahu & Nair (Reference Sumanasiri, Sahu and Nair2025). (b) Phase variable
$\theta (t)$
defined by the pitch angle and its time derivative with the representative flow fields shown. Figures courtesy of the authors.
Phase-reduction analysis is not limited to single-component systems. It can also be applied to multi-physics problems. In the recent paper by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025), the fluid–structure interaction problem of flutter in low-Reynolds-number flow is computationally analysed. They study the occurrence of flutter for a lightweight NACA 0015 aerofoil that is free to pitch about its one-third chord location at a chord-based Reynolds number of 1000, as illustrated in figure 1. By characterising the phase response function (reported as the PSF in their figures 3c and 3d) through impulsive modification of the spring stiffness, they find that the phase dynamics is most receptive to changes between the maximum pitching amplitude and the return to the equilibrium position. They also observe interesting departures of the flutter dynamics from the basin of attraction, as evident from the peaks appearing in the phase response functions. In their work, they make extensive connections between the insights gained from phase reduction and the flow physics by incorporating modal analysis and vortical force analysis.
To prevent the aerofoil pitching dynamics from departing from the basin of attraction, Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) further considered introducing heaving. They quantified the PSF for heaving motion and determined a phase-based control strategy that disrupts the coherence between the structural motion and unsteady aerodynamic forcing. They succeeded with this approach and were able to suppress the violent flutter. This overall suppression of flutter was achieved not only by directly applying linear phase-reduction analysis but also by cleverly leveraging physical insights gained from analysing the nonlinear vortex dynamics around the aerofoil. They have also incorporated energy transfer assessment between the flow and structural dynamics to extend their findings.
Moreover, the work by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) is refreshing in light of the growing interest in using novel structural properties to modify the dynamics of fluid flows around bodies. The impulsive modification of material properties requires additional examination to theoretically quantify its parametric effects on the phase dynamics. Such an examination invites new research initiatives to further advance fluid–structure interaction research. In fact, metamaterials may play a role in fluid–structure interaction in a passive or active manner (Bertoldi et al. Reference Bertoldi, Vitelli, Christensen and van Hecke2017; Coulais, Sounas & Alù Reference Coulais, Sounas and Alù2017). By understanding the sensitivity of the phase dynamics at precise instances in relationship to dynamical events, novel metamaterial-based control strategies as well as structural designs may be developed. The formulation and examination process presented by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) will likely stimulate these new research activities.
3. Summary and outlook
The recent work by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) demonstrated the use of phase-reduction analysis on a fluid–structure interaction problem involving flutter. By combining the insights from phase-reduction analysis with modal analysis, the authors offered deep insights into the mechanism of synchronisation (desynchronisation) and their use in suppressing undesirable flutter oscillations. Their study encourages future applications of similar approaches to complex multi-physics problems in which oscillatory dynamics play a key role.
Compression of high-dimensional dynamical systems to their phase dynamics can also be systematically aided by the use of machine-learning-based techniques. Recently, nonlinear autoencoders have been used to uncover latent-variable dynamics around limit cycles (Fukami, Nakao & Taira Reference Fukami, Nakao and Taira2024; Yawata et al. Reference Yawata, Sakuma, Fukami, Taira and Nakao2025). The discovery of low-dimensional representations through modern data-driven techniques can facilitate the phase-based modelling and control efforts not only for oscillatory flows, but also for transient flows (Fukami et al. Reference Fukami, Nakao and Taira2024). These methods can serve as a foundation to help develop phase-based models for complex multi-frequency turbulent flows.
There are open questions regarding how phase reduction analysis can be applied to turbulent flows with dominant oscillation frequencies. Since such flows do not possess an exact limit cycle, the application of phase reduction analysis requires extensions to quantify the uncertainty in phase and amplitude around the notional cyclic dynamics. This may be achieved by combining phase reduction analysis with modal analysis, such as the dynamic mode decomposition or resolvent analysis, as considered by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025). Furthermore, establishing systematic approaches to analyse energy transfers between oscillatory modes or subsystems can serve as an important tool for studying complex multi-component systems. Developments in phase-based analysis have the potential to offer novel reduced-order models and analysis techniques that can not only provide insights into unsteady physical mechanisms, but also serve as a foundation for aerodynamic and structural design optimisation, as the work by Sumanasiri et al. (Reference Sumanasiri, Sahu and Nair2025) foreshadows.
Acknowledgements
The author is grateful to Hiroya Nakao for the stimulating exchange of ideas on phase-reduction analysis over the years.
Funding
The author thanks the Vannevar Bush Faculty Fellowship from the US Department of Defense (N00014-22-1-2798) for its support.
Declaration of interests
The author reports no conflict of interest.