Dynamical states and bifurcations in coupled thermoacoustic oscillators

High-order synchronization in identical neurons with asymmetric pulse coupling

The phenomenon of high-order (p/q) synchronization, induced by two different frequencies in the system, is well known and studied extensively in forced oscillators including neurons and to a lesser extent in coupled oscillators. Their frequencies are locked such that for every p cycles of one oscillator there are q cycles of the other. We demonstrate this phenomenon in a pair of coupled neurons having identical frequencies but asymmetric coupling. Specifically, we focus on an excitatory-inhibitory (E-I) neuron pair where such an asymmetry is naturally present even with equal reciprocal synaptic strengths (g) and inverse time constant (α). We thoroughly investigate the asymmetric coupling-induced p/q frequency-locking structure in (g,α) parameter space through simulations and analysis. Simulations display quasiperiodicity, devil staircase, a Farey arrangement of spike sequences, and presence of reducible and irreducible p/q regions. We introduce an analytical method, based on event-driven maps, to determine the existence and stability of any spike sequence of the two neurons in a p/q frequency-locked state. Specifically, this method successfully deals with nonsmooth bifurcations and we could utilize it to obtain solutions for the case of identical E-I neuron pair under arbitrary coupling strength. In contrast to the so-called Arnold tongues, the p/q regions obtained here are not structureless. Instead they have their own internal bifurcation structure with varying levels of complexity. Intrasequence and intersequence multistability, involving spike sequences of same p/q state, are found. Additionally, multistability also arises by overlap of p/q with p^{'}/q^{'}. The boundaries of both reducible and irreducible p/q regions are defined by saddle node and nonsmooth grazing bifurcations of various types.

Fokker-Planck dynamics of two coupled van der Pol oscillators subjected to stochastic forcing

We present probabilistic solutions to a pair of mutually coupled Van der Pol oscillators subjected to stochastic forcing. We consider three different types of coupling: reactive, dissipative, and nonlinear coupling. Using stochastic averaging, we derive the stationary Fokker-Planck equation for each oscillator, yielding a probability density function for the fluctuation amplitude. For each coupling type, we numerically validate the Fokker-Planck solutions for different noise levels and coupling strengths, with a focus on the stochastic and bifurcation characteristics. The validated analytical expressions derived in this study could serve to improve the prediction and control of a generic class of coupled oscillator systems operating near the Hopf point in the presence of noise.

Robust design against frequency variation for amplitude death in delay-coupled oscillators

Amplitude death has the potential to suppress unwanted oscillations in various engineering applications. However, in some engineering applications, such as dc microgrids, airfoil systems, and thermoacoustic systems, oscillation frequency is highly susceptible to external influences, leading to considerable variations. To maintain amplitude death amidst these frequency variations, we propose a design procedure that is robust against frequency variation for inducing amplitude death in delay-coupled oscillators. We first analytically derive the oscillator frequency band in which amplitude death can occur. The frequency bandwidth is maximized when the coupling strength is inversely proportional to the connection delay. Furthermore, our analysis reveals that the oscillator frequency band is influenced by the minimum eigenvalue of the normalized adjacency matrix (i.e., network topology) and that bipartite networks exhibit limited robustness to frequency variations. Our design procedure maintains the stability of amplitude death even under substantial frequency variations and is applicable to various network topologies. Numerical simulations confirm the validity of the proposed design.

Delay-induced amplitude death in multiplex oscillator network with frequency-mismatched layers

The present paper analytically investigates the stability of amplitude death in a multiplex Stuart-Landau oscillator network with a delayed interlayer connection. The network consists of two frequency-mismatched layers, and all oscillators in each layer have identical frequencies. We show that, if the matrices describing the network topologies of each layer commute, then the characteristic equation governing the stability can be reduced to a simple form. This form reveals that the stability of amplitude death in the multiplex network is equally or more conservative than that in a pair of frequency-mismatched oscillators coupled by a delayed connection. In addition, we provide a procedure for designing the delayed interlayer connection such that amplitude death is stable for any commuting matrices and for any intralayer coupling strength. These analytical results are verified through numerical examples. Moreover, we numerically discuss the results for the case in which the commutative property does not hold.

Mitigation of limit cycle oscillations in a turbulent thermoacoustic system via delayed acoustic self-feedback

We report the occurrence of amplitude death (AD) of limit cycle oscillations in a bluff body stabilized turbulent combustor through delayed acoustic self-feedback. Such feedback control is achieved by coupling the acoustic field of the combustor to itself through a single coupling tube attached near the anti-node position of the acoustic standing wave. We observe that the amplitude and dominant frequency of the limit cycle oscillations gradually decrease as the length of the coupling tube is increased. Complete suppression (AD) of these oscillations is observed when the length of the coupling tube is nearly 3 / 8 times the wavelength of the fundamental acoustic mode of the combustor. Meanwhile, as we approach this state of amplitude death, the dynamical behavior of acoustic pressure changes from the state of limit cycle oscillations to low-amplitude chaotic oscillations via intermittency. We also study the change in the nature of the coupling between the unsteady flame dynamics and the acoustic field as the length of the coupling tube is increased. We find that the temporal synchrony between these oscillations changes from the state of synchronized periodicity to desynchronized aperiodicity through intermittent synchronization. Furthermore, we reveal that the application of delayed acoustic self-feedback with optimum feedback parameters completely disrupts the positive feedback loop between hydrodynamic, acoustic, and heat release rate fluctuations present in the combustor during thermoacoustic instability, thus mitigating instability. We anticipate this method to be a viable and cost-effective option to mitigate thermoacoustic oscillations in turbulent combustion systems used in practical propulsion and power systems.

Rijke tube: A nonlinear oscillator

Dynamical systems theory has emerged as an interdisciplinary area of research to characterize the complex dynamical transitions in real-world systems. Various nonlinear dynamical phenomena and bifurcations have been discovered over the decades using different reduced-order models of oscillators. Different measures and methodologies have been developed theoretically to detect, control, or suppress the nonlinear oscillations. However, obtaining such phenomena experimentally is often challenging, time-consuming, and risky mainly due to the limited control of certain parameters during experiments. With this review, we aim to introduce a paradigmatic and easily configurable Rijke tube oscillator to the dynamical systems community. The Rijke tube is commonly used by the combustion community as a prototype to investigate the detrimental phenomena of thermoacoustic instability. Recent investigations in such Rijke tubes have utilized various methodologies from dynamical systems theory to better understand the occurrence of thermoacoustic oscillations and their prediction and mitigation, both experimentally and theoretically. The existence of various dynamical behaviors has been reported in single and coupled Rijke tube oscillators. These behaviors include bifurcations, routes to chaos, noise-induced transitions, synchronization, and suppression of oscillations. Various early warning measures have been established to predict thermoacoustic instabilities. Therefore, this review article consolidates the usefulness of a Rijke tube oscillator in terms of experimentally discovering and modeling different nonlinear phenomena observed in physics, thus transcending the boundaries between the physics and the engineering communities.