Stuart-Landau Model Research Articles

Exactly solvable Stuart-Landau models in arbitrary dimensions.

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions D>2 and give an exact solution of the oscillator equationsin the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has N=⌊D/2⌋ pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd D there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere S^{D-1} and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits, each of which has the geometry of a torus T^{N} on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalizations.

How to design optimal brain stimulation to modulate phase-amplitude coupling?

Objective. Phase-amplitude coupling (PAC), the coupling of the amplitude of a faster brain rhythm to the phase of a slower brain rhythm, plays a significant role in brain activity and has been implicated in various neurological disorders. For example, in Parkinson’s disease, PAC between the beta (13–30 Hz) and gamma (30–100 Hz) rhythms in the motor cortex is exaggerated, while in Alzheimer’s disease, PAC between the theta (4–8 Hz) and gamma rhythms is diminished. Modulating PAC (i.e. reducing or enhancing PAC) using brain stimulation could therefore open new therapeutic avenues. However, while it has been previously reported that phase-locked stimulation can increase PAC, it is unclear what the optimal stimulation strategy to modulate PAC might be. Here, we provide a theoretical framework to narrow down the experimental optimisation of stimulation aimed at modulating PAC, which would otherwise rely on trial and error. Approach. We make analytical predictions using a Stuart–Landau model, and confirm these predictions in a more realistic model of coupled neural populations. Main results. Our framework specifies the critical Fourier coefficients of the stimulation waveform which should be tuned to optimally modulate PAC. Depending on the characteristics of the amplitude response curve of the fast population, these components may include the slow frequency, the fast frequency, combinations of these, as well as their harmonics. We also show that the optimal balance of energy between these Fourier components depends on the relative strength of the endogenous slow and fast rhythms, and that the alignment of fast components with the fast rhythm should change throughout the slow cycle. Furthermore, we identify the conditions requiring to phase-lock stimulation to the fast and/or slow rhythms. Significance. Together, our theoretical framework lays the foundation for guiding the development of innovative and more effective brain stimulation aimed at modulating PAC for therapeutic benefit.

Empirical Modeling of Subcritical Hopf Bifurcation of Thermoacoustic Stirling Engine

This study models the subcritical Hopf bifurcation in thermoacoustic Stirling engines using the Stuart–Landau model, highlighting the role of nonlinear dynamics. By inducing self-sustained oscillations and measuring pressure fluctuations across different temperature gradients imposed on the regenerator, we reveal the engine’s transition to a nonlinear domain, characterized by heightened oscillation amplitudes and unique periodic patterns. Interpreted Landau constants and growth rates illuminate the stabilizing effects of nonlinear dynamics, demonstrating the Stuart–Landau model’s applicability in thermoacoustic engine analysis. Our research confirms that this empirically refined model reliably describes oscillation amplitudes and transient phenomena, contributing valuable perspectives for advancing thermoacoustic engine design and operational understanding.

Synchronization modes of triple flickering buoyant diffusion flames: Experimental identification and model interpretation.

The synchronization modes of a nonlinear oscillator system consisting of three identical flickering buoyant diffusion flames in isosceles triangles were studied experimentally and theoretically. Five synchronization modes, such as the in-phase, flickering death, partially flickering death, partially in-phase, and rotation modes, were experimentally observed and identified by systematically adjusting the flame distance and fuel flow rates. Two toy models were adopted to interpret the experimentally identified dynamical modes: one is the classical Kuramoto model, and the other is a complexified Stuart-Landau model, which was proposed through the introduction of the complex coupling term. The theoretical results show that the Kuramoto model successfully interpreted the dynamical modes except for those associated with amplitude death, and the complexified Stuart-Landau model well interpreted all the dynamical modes identified in our experiment. Remarkably, the proposed complexified Stuart-Landau model breaks a new path in the investigation of globally coupled nonlinear dynamical systems with identical oscillators, especially for the study of amplitude death mode.

Weakly nonlinear bifurcation behaviour at the onset of instability in horizontal convection using an advective buoyancy approximation

Horizontal free convection is studied under an extended buoyancy approximation constructed to capture centrifugal effects in strongly rotating regions within buoyancy driven flows. Under this approximation, the Gay-Lussac parameter (Ga) arises in the momentum equation to characterise non-Boussinesq behaviour. The scaling of average Nusselt numbers (Nuavg) are determined, as is its dependence on Ga. Increasing Ga is found to decrease Nuavg when convection dominates the flow, while no influence is detected in the low Rayleigh number regime dominated by conduction. The influence of Ga on the onset of the time-dependent regime is investigated and the critical Rayleigh number corresponding to different Ga are determined. Results demonstrate that increasing Ga stabilises the flow, delaying transition to the time-dependent regime. An Orr–Sommerfeld type stability analysis is performed to determine the local stability at different horizontal stations. A zone of convective instability is first detected, growing from the hot end of the enclosure, at Rayleigh numbers three orders of magnitude lower than the predicted global stability threshold at the maximum investigated Ga value. Ga is found to play a key role in determining the preferred orientation of instability roll structures, with increasing Ga being accompanied by a bias toward transverse-roll instabilities over longitudinal rolls. The Stuart—Landau model is used to reveal that the non-linear characteristics of this unsteady transition are consistent with a supercritical Hopf bifurcation.

Modeling of Main Parameters for Near Wake from Pair of Side-by-Side not too Close Cylinders

A one-dimensional model of the near wake produced by a pair of not-too-close cylinders in a viscous incompressible fluid is developed on the basis of the Stuart–Landau model and perturbation theory. The whole wake is considered as two interacting partial von Kàrmàn vortex streets or two coupled Stuart–Landau oscillators, when the nonlinear nature of the interaction of these streets is taken into account. The observable spectrum of the global modes of the wake is obtained, the calculated and experimental eigenfrequencies of the oscillation modes being in agreement with respect to the modes.

The impact of regional heterogeneity in whole-brain dynamics in the presence of oscillations.

Large variability exists across brain regions in health and disease, considering their cellular and molecular composition, connectivity, and function. Large-scale whole-brain models comprising coupled brain regions provide insights into the underlying dynamics that shape complex patterns of spontaneous brain activity. In particular, biophysically grounded mean-field whole-brain models in the asynchronous regime were used to demonstrate the dynamical consequences of including regional variability. Nevertheless, the role of heterogeneities when brain dynamics are supported by synchronous oscillating state, which is a ubiquitous phenomenon in brain, remains poorly understood. Here, we implemented two models capable of presenting oscillatory behavior with different levels of abstraction: a phenomenological Stuart-Landau model and an exact mean-field model. The fit of these models informed by structural- to functional-weighted MRI signal (T1w/T2w) allowed us to explore the implication of the inclusion of heterogeneities for modeling resting-state fMRI recordings from healthy participants. We found that disease-specific regional functional heterogeneity imposed dynamical consequences within the oscillatory regime in fMRI recordings from neurodegeneration with specific impacts on brain atrophy/structure (Alzheimer's patients). Overall, we found that models with oscillations perform better when structural and functional regional heterogeneities are considered, showing that phenomenological and biophysical models behave similarly at the brink of the Hopf bifurcation.

Scale-free dynamics in the core-periphery topography and task alignment decline from conscious to unconscious states

Scale-free physiological processes are ubiquitous in the human organism. Resting-state functional MRI studies observed the loss of scale-free dynamics under anesthesia. In contrast, the modulation of scale-free dynamics during task-related activity remains an open question. We investigate scale-free dynamics in the cerebral cortex’s unimodal periphery and transmodal core topography in rest and task states during three conscious levels (awake, sedation, and anesthesia) complemented by computational modelling (Stuart-Landau model). The empirical findings demonstrate that the loss of the brain’s intrinsic scale-free dynamics in the core-periphery topography during anesthesia, where pink noise transforms into white noise, disrupts the brain’s neuronal alignment with the task’s temporal structure. The computational model shows that the stimuli’s scale-free dynamics, namely pink noise distinguishes from brown and white noise, also modulate task-related activity. Together, we provide evidence for two mechanisms of consciousness, temporo-spatial nestedness and alignment, suggested by the Temporo-Spatial Theory of Consciousness (TTC).

Dynamic Survivability in Nonlinear Oscillation Systems with Attractive–Repulsive Interaction

On the basis of global and BA scale-free coupled Stuart–Landau models, dynamic survivability has been investigated in detail with new definition and measure function, which is different from the previous survivability studies which only focused on static analysis. The effects on dynamic survivability of attractive–repulsive interaction and attack strategies are detected. Our results suggest that the coupling strength and presence of the repulsive interaction reduce the dynamic survivability in globally coupled systems. Furthermore, the dynamic survivability of the BA systems remains stable in the case of random attack with invariable critical attack cost [Formula: see text]. While they have the same features with globally coupled systems when being deliberately attacked; attacking high-degree oscillators show a tendency to spoil the dynamic survivability more effectively. Finally, it is found that the attractive coupling plays a more important role in the dynamic survivability. These findings may help us to prevent systems from being attacked and design survivable systems.

Effect of Base Fluids on the Initiation of Vortex Shedding for Nanofluid Flow Over a Circular Object

Abstract An inherently steady separated flow of a nanofluid around a two-dimensional circular object can be transformed into an unsteady periodic flow by controlling the nanoparticle concentration. This has been established in the contemporary literature 2022 by Garai et al. (2022, “Triggering Vortex Shedding for the Free Stream Flow of Nanofluids Around Bluff Objects,” ASME J. Fluids Eng., 144(3), p. 034502.) using Cu–H2O and Ag–H2O nanofluids, where the base fluids remain the same (H2O) and the nanoparticles are different (Cu, Ag). However, it may be anticipated that changing the base fluid may cause certain alterations in the flow dynamics. Two different base fluids, e.g., propylene glycol (C3H8O2) and ethylene glycol (C2H6O2) are selected in which copper (Cu) nanoparticles are mixed to produce Cu–C3H8O2 and Cu–C2H6O2 nanofluids. The nanofluids flow over a circular object with Reynolds number in the range 15≤Re≤40. The Cu nanoparticle concentration is gradually increased and it is observed that the vortex shedding can be initiated with a lesser concentration of Cu in case of C3H8O2 based nanofluid compared to C2H6O2 and also H2O based nanofluids. Hence, the critical solid volume fractions (φcr) for Cu–C2H6O2 based nanofluid are more compared to Cu–C3H8O2 and Cu–H2O nanofluids. Further, φcr for all the nanofluids are found a decreasing function of Re in its chosen range. The critical solid fraction is estimated from an extended Stuart–Landau model and the phenomena are demonstrated qualitatively through the streamlines and vorticity contours and quantitatively through the lift signal analysis and phase diagrams. Finally, a regime diagram is constructed to demarcate the steady and unsteady zones of operation.

Frequency-amplitude correlation inducing first-order phase transition in coupled oscillators

The first-order phase transitions in coupled oscillators have been widely studied because of their discontinuity and irreversibility. In previous research, the designed coupling mechanisms between each pair of oscillators can cause the first-order phase transitions occur stably. In the present study, we propose a new mechanism which requires the existence of an inversely proportional relationship between the natural frequencies and the intrinsic amplitudes in the homogeneously coupled oscillators. Based on two classical oscillator models, i.e., the Poincaré model and the Stuart–Landau model, the emergence of explosive oscillation death is independent of the frequency distributions. Our findings indicate that the first-order phase transitions can be induced by the frequency-amplitude correlation for the first time. Therefore, it provides a novel perspective to understand explosive phenomena in coupled oscillators.

Duffing–van der Pol nonlinear reduced-order model for explaining the phenomena and mechanism in pulsed jet flow separation control

A two-equation nonlinear reduced-order model is established to describe the interaction between external periodic excitation from a pulsed jet and unsteady flow separation. This model, which is based on a 2D Navier–Stokes equation, the Stuart vortex row model, and the Stuart–Landau model, can approximately describe the typical phenomena occurring in pulsed jet flow separation control. These phenomena include the frequency-dependent, threshold, and lock-on effects, which are related to the frequency, intensity, and phase of excitation. Two indices, namely, the maximum Lyapunov exponent and the entrainment degree, are introduced to identify the mechanism of flow control via external periodic excitation. These indices are used to reflect the order degree and momentum transfer of the flow field. A comparison of the results of the model and those of a numerical simulation in a curved diffuser shows that the nonlinear reduced-order model is effective for qualitatively featuring the behavior of pulsed jet flow separation control. Moreover, the mechanism of pulsed jet flow separation control is explained via the model. Three aspects (i.e., flow instability, flow field ordering, and momentum transfer) are assumed to function together, resulting in an efficient flow control performance.

Amplitude-phase description of stochastic neural oscillators across the Hopf bifurcation

We derive a unified amplitude-phase decomposition for both noisy limit cycles and quasicycles; in the latter case, the oscillatory motion has no deterministic counterpart. We extend a previous amplitude-phase decomposition approach using the stochastic averaging method (SAM) for quasicycles by taking into account nonlinear terms up to order 3. We further take into account the case of coupled networks where each isolated network can be in a quasi- or noisy limit-cycle regime. The method is illustrated on two models which exhibit a deterministic supercritical Hopf bifurcation: the Stochastic Wilson-Cowan model of neural rhythms, and the Stochastic Stuart-Landau model in physics. At the level of a single oscillatory module, the amplitude process of each of these models decouples from the phase process to the lowest order, allowing a Fokker-Planck estimate of the amplitude probability density. The peak of this density captures well the transition between the two regimes. The model describes accurately the effect of Gaussian white noise as well as of correlated noise. Bursting epochs in the limit-cycle regime are in fact favored by noise with shorter correlation time or stronger intensity. Quasicycle and noisy limit-cycle dynamics are associated with, respectively, Rayleigh-type and Gaussian-like amplitude densities. This provides an additional tool to distinguish quasicycle from limit-cycle origins of bursty rhythms. The case of multiple oscillatory modules with excitatory all-to-all delayed coupling results in a system of stochastic coupled amplitude-phase equations that keeps all the biophysical parameters of the initial networks and again works across the Hopf bifurcation. The theory is illustrated for small heterogeneous networks of oscillatory modules. Numerical simulations of the amplitude-phase dynamics obtained through the SAM are in good agreement with those of the original oscillatory networks. In the deterministic and nearly identical oscillators limits, the stochastic Stuart-Landau model leads to the stochastic Kuramoto model of interacting phases. The approach can be tailored to networks with different frequency, topology, and stochastic inputs, thus providing a general and flexible framework to analyze noisy oscillations continuously across the underlying deterministic bifurcation.

Antiresonance in switched systems with only unstable modes

The authors report the emergence of antiresonance in a variation of the Stuart-Landau model, which alternates between two unstable modes but displays a stable response in a finite range of switching frequencies.

Effect of amplitude and frequency of limit cycle oscillators on their coupled and forced dynamics

The occurrence of synchronization and amplitude death phenomena due to the coupled interaction of limit cycle oscillators (LCO) has received increased attention over the last few decades in various fields of science and engineering. Studies pertaining to these coupled oscillators are often performed by studying the effect of various coupling parameters on their mutual interaction. However, the effect of system parameters (i.e., the amplitude and frequency) on the coupled interaction of such LCO has not yet received much attention, despite their practical importance. In this paper, we investigate the dynamical behavior of time-delay coupled Stuart–Landau (SL) oscillators exhibiting subcritical Hopf bifurcation for the variation of amplitude and frequency of these oscillators in their uncoupled state. For identical SL oscillators, a gradual increase in the amplitude of LCO shrinks the amplitude death regions observed between the regions of in-phase and anti-phase synchronization leading to its eventual disappearance, resulting in the occurrence of phase-flip bifurcations at higher amplitudes of LCO. We also observe an alternate existence of in-phase and anti-phase synchronization regions for higher values of time delay, whose prevalence of occurrence increases with an increase in the frequency of the oscillator. With the introduction of frequency mismatch, the region of amplitude death. The forced response of SL oscillator shows an asymmetry in the Arnold tongue and the manifestation of asynchronous quenching of LCO. An increase in the amplitude of LCO narrows the Arnold tongue and reduces the region of asynchronous quenching observed in the system. Finally, we compare the coupled and forced response of SL oscillators with the corresponding experimental results obtained from laminar thermoacoustic oscillators and the numerical results from van der Pol (VDP) oscillators. We show that the SL model qualitatively displays many features observed experimentally in coupled and forced thermoacoustic oscillators. In contrast, the VDP model does not capture most of the experimental results due to the limitation in the independent variation of system parameters.

Modeling synchronization in forced turbulent oscillator flows

Periodically forced, oscillatory fluid flows have been the focus of intense research for decades due to their richness as a nonlinear dynamical system and their relevance to applications in transportation, aeronautics, and energy conversion. Here we derive a mechanistic model of the dynamics of forced turbulent oscillator flows by leveraging a comprehensive experimental study of the turbulent wake behind a D-shaped body under periodic forcing. We confirm the role of resonant triadic interactions in the forced flow by studying the dominant components in the power spectra across multiple excitation frequencies and amplitudes. We then develop an extended Stuart-Landau model that captures the system dynamics and synchronization regions. Further, it is possible to identify the model coefficients from sparse measurement data.

EEG microstate periodicity explained by rotating phase patterns of resting-state alpha oscillations

Spatio-temporal patterns in electroencephalography (EEG) can be described by microstate analysis, a discrete approximation of the continuous electric field patterns produced by the cerebral cortex. Resting-state EEG microstates are largely determined by alpha frequencies (8-12 Hz) and we recently demonstrated that microstates occur periodically with twice the alpha frequency.To understand the origin of microstate periodicity, we analyzed the analytic amplitude and the analytic phase of resting-state alpha oscillations independently. In continuous EEG data we found rotating phase patterns organized around a small number of phase singularities which varied in number and location. The spatial rotation of phase patterns occurred with the underlying alpha frequency. Phase rotors coincided with periodic microstate motifs involving the four canonical microstate maps. The analytic amplitude showed no oscillatory behaviour and was almost static across time intervals of 1-2 alpha cycles, resulting in the global pattern of a standing wave.In n=23 healthy adults, time-lagged mutual information analysis of microstate sequences derived from amplitude and phase signals of awake eyes-closed EEG records showed that only the phase component contributed to the periodicity of microstate sequences. Phase sequences showed mutual information peaks at multiples of 50 ms and the group average had a main peak at 100 ms (10 Hz), whereas amplitude sequences had a slow and monotonous information decay. This result was confirmed by an independent approach combining temporal principal component analysis (tPCA) and autocorrelation analysis.We reproduced our observations in a generic model of EEG oscillations composed of coupled non-linear oscillators (Stuart-Landau model). Phase-amplitude dynamics similar to experimental EEG occurred when the oscillators underwent a supercritical Hopf bifurcation, a common feature of many computational models of the alpha rhythm.These findings explain our previous description of periodic microstate recurrence and its relation to the time scale of alpha oscillations. Moreover, our results corroborate the predictions of computational models and connect experimentally observed EEG patterns to properties of critical oscillator networks.

A Study of Rayleigh-Bénard-Taylor Convection in Very-Shallow, Shallow, Square and Tall Enclosures

The paper deals with an analytical-study of Rayleigh-Benard-convection (RBC) within four types of enclosures in a non-inertial frame of reference(NIFR) due to uniform rotation of the system. Using a double Fourier trigonometric series with least number of modes, an extended Lorenz model which has in it the effect of an NIFR is derived under the assumption of the Oberbeck-Boussinesq (OB) approximation. The method of multi-scales is employed to arrive at the first-order, autonomous Stuart-Landau equation. Its solution is analysed and it is shown that though the Stuart-Landau model is lower in order than the Lorenz model it retains some of the features of the Lorenz model and, further, facilitates an analytical solution. Using this solution, the expression of the Nusselt number is obtained in closed form. The study reveals that the NIFR-effect leads to a stabilized-system and a decrease in the amount of heat-transport. The rate of heat-transport-diminishment is discussed using an Ozoe coefficient. Besides these findings, the study reveals that from the results of the classical problem of RBC in an NIFR the results of the other three enclosures may be obtained by suitable redefinition of the parameters.

Overcoming Oscillation Quenching via Mean-Field Feedback

Observational studies have shown that many natural systems need to remain in rhythmic oscillation activity. Yet, the appearance of oscillation quenching phenomena lead to the breakdown of intrinsic behavior of practical systems, often causing unpredictable results. To address the important problem about how to avoid or revive the paralyzed system, in this work, we use the mean-field function of the entire system as an external feedback acts on every unit. Taking the classical limit cycle of Stuart–Landau model as example, we find that oscillation quenching state can be eliminated effectively and thus revoke the deprived oscillation. Through extensive theoretical analyses and numerical simulations, the results show for various death scenarios that the oscillation quenching regions shrink drastically in the parameter space with increasing feedback strength. Our method can be regarded as a general framework to maintain or retrieve the continued oscillation for coupled nonlinear systems in diverse applications.

Abstract #4392 Correlative effects of cancer-related fatigue on the neuroendocrine-immune network: Role of hormone responsiveness, metastatic markers, immune markers and psychosocial factors

Spatio-temporal patterns in electroencephalography (EEG) can be described by microstate analysis, a discrete approximation of the continuous electric field patterns produced by the cerebral cortex. Resting-state EEG microstates are largely determined by alpha frequencies (8-12 Hz) and we recently demonstrated that microstates occur periodically with twice the alpha frequency.To understand the origin of microstate periodicity, we analyzed the analytic amplitude and the analytic phase of resting-state alpha oscillations independently. In continuous EEG data we found rotating phase patterns organized around a small number of phase singularities which varied in number and location. The spatial rotation of phase patterns occurred with the underlying alpha frequency. Phase rotors coincided with periodic microstate motifs involving the four canonical microstate maps. The analytic amplitude showed no oscillatory behaviour and was almost static across time intervals of 1-2 alpha cycles, resulting in the global pattern of a standing wave.In n=23 healthy adults, time-lagged mutual information analysis of microstate sequences derived from amplitude and phase signals of awake eyes-closed EEG records showed that only the phase component contributed to the periodicity of microstate sequences. Phase sequences showed mutual information peaks at multiples of 50 ms and the group average had a main peak at 100 ms (10 Hz), whereas amplitude sequences had a slow and monotonous information decay. This result was confirmed by an independent approach combining temporal principal component analysis (tPCA) and autocorrelation analysis.We reproduced our observations in a generic model of EEG oscillations composed of coupled non-linear oscillators (Stuart-Landau model). Phase-amplitude dynamics similar to experimental EEG occurred when the oscillators underwent a supercritical Hopf bifurcation, a common feature of many computational models of the alpha rhythm.These findings explain our previous description of periodic microstate recurrence and its relation to the time scale of alpha oscillations. Moreover, our results corroborate the predictions of computational models and connect experimentally observed EEG patterns to properties of critical oscillator networks.