Phase Reduction Approach Research Articles

Optimal waveform for fast synchronization of airfoil wakes

We obtain an optimal actuation waveform for fast synchronization of periodic airfoil wakes through the phase reduction approach. Using the phase reduction approach for periodic wake flows, the spatial sensitivity fields with respect to the phase of the vortex shedding are obtained. The phase sensitivity fields can uncover the synchronization properties in the presence of periodic actuation. This study seeks a periodic actuation waveform using phase-based analysis to minimize the time for synchronization to modify the wake shedding frequency of NACA0012 airfoil wakes. This fast synchronization waveform is obtained theoretically from the phase sensitivity function by casting an optimization problem. The obtained optimal actuation waveform becomes increasingly non-sinusoidal for higher angles of attack. Actuation based on the obtained waveform achieves rapid synchronization within as low as two vortex shedding cycles irrespective of the forcing frequency, whereas traditional sinusoidal actuation requires ${O}(10)$ shedding cycles. Further, we analyse the influence of actuation frequency on the vortex shedding and the aerodynamic coefficients using force-element analysis. The present analysis provides an efficient way to modify the vortex lock-on properties in a transient manner with applications to fluid–structure interactions and unsteady flow control.

Embedding phase reduction for fast-slow systems with noise-induced stochastic quasiperiodic orbits

Noise is ubiquitous in natural systems and can induce coherent stochastic oscillations which cannot occur in the absence of noise, or stochastic quasiperiodic orbits which are markedly different from the deterministic limit cycle. For fast-slow systems, noise acting on the fast subsystem can induce quasi-periodic orbits tuned by the slow variable, which were termed as self-induced stochastic resonance (SISR). In our previous work [Zhu et al., 2022 [46]], we proposed a phase reduction framework on the SISR-oscillator by the hybrid approximation on the stochastic quasiperiodic orbit and the estimation of the effective noise. In this work, we remove the restriction of the additive Gaussian form assumption on the effective noise, and propose a more natural dimensionality-reduction method which we refer to as embedding phase reduction. Different SISR-oscillators induced by varying noise intensities, which can be embedded into a limit cycle, can have a unified form of the reduced phase equation. The efficiency of the embedding phase reduction is validated by Monte Carlo simulations of the case of periodic forcing, where excellent agreement is achieved for the entrainment and phase drifting dynamics. The simplicity and generality of the proposed embedding phase reduction approach indicate its potential applications in practical experiments and relevance in biological systems.

Spatial Effects of Phase Dynamics on Oscillators Close to Bifurcation

The phase reduction approach has manifested its efficacy in investigating synchronization behaviors in limit-cycle oscillators. However, spatial distributions of the phase value on the limit cycle may lead to illusions of synchronizations for oscillators close to bifurcations. In this paper, we compared the phase sensitivity function in the spatial domain and time domain for oscillators close to saddle-node homoclinic (SNH) bifurcation, also known as saddle-node bifurcation on an invariant circle. It was found that the phase sensitivity function in the spatial domain can show the phase accumulation feature on the limit cycle, which can be ignored in the phase sensitivity function in the time domain. As an example, the synchronization distributions of uncoupled SNH oscillators driven by common and independent noises were investigated, where the space-dependent coupling function was considered on common noise. These results shed some light on the phase dynamics of oscillators close to bifurcations.

Highly efficient removal of ionic dyes in aqueous solutions using magnetic 3D reduced graphene oxide aerogel supported nano zero-valent iron

This study aims to develop three-dimension reduced graphene oxide aerogel supported nano zero-valent iron (rGOA-nZVI) by hydrothermal and liquid phase reduction approach and applies it to the removal of two typical dyes, methylene blue (MB) and methyl orange (MO) from aqueous solutions. The obtained magnetic rGOA-nZVI composites were characterized by scanning electron microscopy (SEM), atomic force microscopy, Brunauer-Emmett-Teller, X-ray diffraction and Fourier transform infrared spectroscopy, and its removal efficiencies for MB and MO from aqueous solutions were evaluated. The removal processes of MB and MO could be completed within 30 min, which were well described by the pseudo-second order kinetic model. The rGOA-nZVI nanocomposites exhibited remarkable performance for the removal of MB (the maximum adsorption amount, <i>q</i><sub>max</sub> = 3918 mg/g) and MO (<i>q</i><sub>max</sub> = 667 mg/g) in a wide pH range (pH 1.0-10.0). Furthermore, rGOA-nZVI is stable and recyclable with high dye removal efficiencies (> 90%) after five cycles. The removal mechanisms were demonstrated with SEM and XRD analyses before and after MB and MO removal. This study showed that rGOA acts as an excellent carrier for nZVI, and the rGOA-nZVI nanocomposites are efficient materials for the advanced treatment of dye wastewater over a wide pH range.

Ni nanospheres coated with tunable carbon shells for dielectric-controlled superior electromagnetic wave absorbing

To obtain high-efficiency electromagnetic wave absorbing (EMA) material with strong reflection loss ( RL ) and wide effective absorbing bandwidth (EAB), Ni particles are successfully prepared in this work by a liquid phase reduction approach, and Ni@C is subsequently designed and fabricated through combined polymerization and carbonization process. The pristine Ni exhibits outstanding EMA performances in middle- and high-frequency with a maximum RL ( RL max ) of 62.3 dB (2.3 mm) at K u band and an EAB of 10.9 GHz (5.6–16.5 GHz) at 3.0 mm, covering 75% K u (12–18 GHz), 100% X (8–12 GHz) and 60% C (4–8 GHz) band. Such a superior property is attributed to the dielectric losses derived from interface polarization, dipole polarization and conduction loss, along with magnetic losses comes from natural and exchange resonance. An introducing of carbon shell with manipulated thicknesses induces positive enhancement in dielectric losses. Meanwhile, core-shell structures enable multiple scattering and reflection of microwaves. Thus, Ni@C spheres exhibit significantly enhanced EMA performances in low-frequency covering S (2–4 GHz) and C band with an RL max at 4.4 GHz reaching 64.4 dB (Ni@C-0.4) and an RL max at 2.6 GHz reaching 59.3 dB (Ni@C-0.6). Therefore, Ni and Ni@C demonstrate a unique blend of favorable EMA performances, which are promising for next-generation intelligent EMA systems.

Phase sensitivity for coherence resonance oscillators

The phase reduction approach has manifested its power in analyzing the rhythmic behaviors for limit cycle oscillators. For coherent oscillation purely induced by noise, e.g., the coherence resonance oscillator, the stochastic dynamics exhibit almost deterministic limit cycle phenomenon which inspires the application of the phase reduction approach to this kind of systems. In this paper, the FitzHugh–Nagumo system in coherence resonance is modeled as a jump process. The phase sensitivity is obtained by applying the phase reduction approach for the hybrid system. A modified direct method is proposed to compare the theoretical results with those of Monte Carlo simulation for the stochastic system, which shows a relatively good agreement for the perturbation not being too small. The phase reduction results of the coherent oscillators are applied to two coupled FitzHugh–Nagumo neurons. It is interesting that the phase difference of the coupled coherent oscillators does not converge as the deterministic oscillators, but forms a distribution where the peaks and valleys correspond to the stable and unstable synchronizations predicted by the phase coupling functions. The idea in this paper could be applied to other coherent oscillators where the stochastic dynamics follow almost deterministic paths.

Phase reduction for FitzHugh-Nagumo neuron with large timescale separation.

Phase reduction approach is a useful tool to reduce the dimension of limit-cycle oscillators. In this paper, it is applied to the FitzHugh-Nagumo neuron model with large timescale separation. To quantitate the response of the oscillator to the external perturbations, the asymptotic behaviors of the phase sensitivity functions for fast and slow variables are obtained. It is shown that in the relaxation limit, apart from the jump points which are the vertices of the cubic nullcline, the phase is insensitive to the perturbations on the fast variable. By using the phase sensitivity functions, two cases, namely, periodic pulse train perturbation and common noise perturbation, are investigated. Theoretical and numerical results suggest better performance for the synchronization behaviors of the perturbations on the slow variable for large timescale separation.

In-phase synchronization in complex oscillator networks by adaptive delayed feedback control

In-phase synchronization is a special case of synchronous behavior when coupled oscillators have the same phases for any time moments. Such behavior appears naturally for nearly identical coupled limit-cycle oscillators when the coupling strength is greatly above the synchronization threshold. We investigate the general class of nearly identical complex oscillators connected into network in a context of a phase reduction approach. By treating each oscillator as a black-box possessing a single-input single-output, we provide a practical and simply realizable control algorithm to attain the in-phase synchrony of the network. For a general diffusive-type coupling law and any value of a coupling strength (even greatly below the synchronization threshold) the delayed feedback control with a specially adjusted time-delays can provide in-phase synchronization. Such adjustment of the delay times performed in an automatic fashion by the use of an adaptive version of the delayed feedback algorithm when time-delays become time-dependent slowly varying control parameters. Analytical results show that there are many arrangements of the time-delays for the in-phase synchronization, therefore we supplement the algorithm by an additional requirement to choose appropriate set of the time-delays, which minimize power of a control force. Performed numerical validations of the predictions highlights the usefulness of our approach.

Phase reduction approach to elastohydrodynamic synchronization of beating flagella.

We formulate a theory for the phase reduction of a beating flagellum. The theory enables us to describe the dynamics of a beating flagellum in a systematic manner using a single variable called the phase. The theory can also be considered as a phase reduction method for the limit-cycle solutions in infinite-dimensional dynamical systems, namely, the limit-cycle solutions to partial differential equations representing beating flagella. We derive the phase sensitivity function, which quantifies the phase response of a beating flagellum to weak perturbations applied at each point and at each time. Using the phase sensitivity function, we analyze the phase synchronization between a pair of beating flagella through hydrodynamic interactions at a low Reynolds number.

Desynchronization of stochastically synchronized chemical oscillators.

Experimental and theoretical studies are presented on the design of perturbations that enhance desynchronization in populations of oscillators that are synchronized by periodic entrainment. A phase reduction approach is used to determine optimal perturbation timing based upon experimentally measured phase response curves. The effectiveness of the perturbation waveforms is tested experimentally in populations of periodically and stochastically synchronized chemical oscillators. The relevance of the approach to therapeutic methods for disrupting phase coherence in groups of stochastically synchronized neuronal oscillators is discussed.

Phase reduction approach to synchronisation of nonlinear oscillators

Systems of dynamical elements exhibiting spontaneous rhythms are found in various fields of science and engineering, including physics, chemistry, biology, physiology, and mechanical and electrical engineering. Such dynamical elements are often modelled as nonlinear limit-cycle oscillators. In this article, we briefly review phase reduction theory, which is a simple and powerful method for analysing the synchronisation properties of limit-cycle oscillators exhibiting rhythmic dynamics. Through phase reduction theory, we can systematically simplify the nonlinear multi-dimensional differential equations describing a limit-cycle oscillator to a one-dimensional phase equation, which is much easier to analyse. Classical applications of this theory, i.e. the phase locking of an oscillator to a periodic external forcing and the mutual synchronisation of interacting oscillators, are explained. Further, more recent applications of this theory to the synchronisation of non-interacting oscillators induced by common noise and the dynamics of coupled oscillators on complex networks are discussed. We also comment on some recent advances in phase reduction theory for noise-driven oscillators and rhythmic spatiotemporal patterns.

Delayed feedback control of synchronization in weakly coupled oscillator networks

We study control of synchronization in weakly coupled oscillator networks by using a phase-reduction approach. Starting from a general class of limit-cycle oscillators we derive a phase model, which shows that delayed feedback control changes effective coupling strengths and effective frequencies. We derive the analytical condition for critical control gain, where the phase dynamics of the oscillator becomes extremely sensitive to any perturbations. As a result the network can attain phase synchronization even if the natural interoscillatory couplings are small. In addition, we demonstrate that delayed feedback control can disrupt the coherent phase dynamic in synchronized networks. The validity of our results is illustrated on networks of diffusively coupled Stuart-Landau and FitzHugh-Nagumo models.

Phase-Amplitude Response Functions for Transient-State Stimuli

The phase response curve (PRC) is a powerful tool to study the effect of a perturbation on the phase of an oscillator, assuming that all the dynamics can be explained by the phase variable. However, factors like the rate of convergence to the oscillator, strong forcing or high stimulation frequency may invalidate the above assumption and raise the question of how is the phase variation away from an attractor. The concept of isochrons turns out to be crucial to answer this question; from it, we have built up Phase Response Functions (PRF) and, in the present paper, we complete the extension of advancement functions to the transient states by defining the Amplitude Response Function (ARF) to control changes in the transversal variables. Based on the knowledge of both the PRF and the ARF, we study the case of a pulse-train stimulus, and compare the predictions given by the PRC-approach (a 1D map) to those given by the PRF-ARF-approach (a 2D map); we observe differences up to two orders of magnitude in favor of the 2D predictions, especially when the stimulation frequency is high or the strength of the stimulus is large. We also explore the role of hyperbolicity of the limit cycle as well as geometric aspects of the isochrons. Summing up, we aim at enlightening the contribution of transient effects in predicting the phase response and showing the limits of the phase reduction approach to prevent from falling into wrong predictions in synchronization problems.List of AbbreviationsPRC phase response curve, phase resetting curve.PRF phase response function.ARF amplitude response function.