Synchronization and chimeras in a network of four ring-coupled thermoacoustic oscillators

Abstract

We take a complex systems approach to investigating experimentally the collective dynamics of a network of four self-excited thermoacoustic oscillators coupled in a ring. Using synchronization metrics, we find a wide variety of emergent multi-scale behaviour, such as (i) a transition from intermittent frequency locking on a $\mathbb {T}^{3}$ quasiperiodic attractor to a breathing chimera, (ii) a two-cluster state of anti-phase synchronization on a periodic limit cycle, and (iii) a weak anti-phase chimera. We then compute the cross-transitivity from recurrence networks to identify the dominant direction of the coupling between the heat-release-rate ( $q^{\prime }_{\mathbb {X}}$ ) and pressure ( $p^{\prime }_{\mathbb {X}}$ ) fluctuations in each individual oscillator, as well as that between the pressure ( $p^{\prime }_{\mathbb {X}}$ and $p^{\prime }_{\mathbb {Y}}$ ) fluctuations in each pair of coupled oscillators. We find that networks of non-identical oscillators exhibit circumferentially biased $p^{\prime }_{\mathbb {X}}$ – $p^{\prime }_{\mathbb {Y}}$ coupling, leading to mode localization, whereas networks of identical oscillators exhibit globally symmetric $p^{\prime }_{\mathbb {X}}$ – $p^{\prime }_{\mathbb {Y}}$ coupling. In both types of networks, we find that the $p^{\prime }_{\mathbb {X}}$ – $q^{\prime }_{\mathbb {X}}$ coupling can be symmetric or asymmetric, but that the asymmetry is always such that $q^{\prime }_{\mathbb {X}}$ exerts a greater influence on $p^{\prime }_{\mathbb {X}}$ than vice versa. Finally, we show through a cluster analysis that the $p^{\prime }_{\mathbb {X}}$ – $p^{\prime }_{\mathbb {Y}}$ interactions play a more critical role than the $p^{\prime }_{\mathbb {X}}$ – $q^{\prime }_{\mathbb {X}}$ interactions in defining the collective dynamics of the system. As well as providing new insight into the interplay between the $p^\prime_{\mathbb{X}}\text{--}p^\prime_{\mathbb{Y}}$ and $p^\prime_{\mathbb{X}}\text{--}q^\prime_{\mathbb{X}}$ coupling, this study shows that even a small network of four ring-coupled thermoacoustic oscillators can exhibit a wide variety of collective dynamics. In particular, we present the first evidence of chimera states in a minimal network of coupled thermoacoustic oscillators, paving the way for the application of oscillation quenching strategies based on chimera control.

Weakly nonlinear analysis of thermoacoustic oscillations in can-annular combustors

“…When the combustors were operated at identical conditions, in-phase and phase-opposition synchronised states occurred sporadically in a competitive manner, with asynchronous states existing at low coupling strength. Guan et al (2022) consider a system composed of four thermoacoustic oscillators and identify synchronisation and other nonlinear phenomena in symmetric and asymmetric settings. (II) Alternatively, one can consider the thermoacoustic modes of the full can-annular system as global oscillatory states that may synchronise.…”

Complex-network analysis of high-frequency combustion instability in a model single-element rocket engine combustor

“…A machine learning approach based on statistical learning theory (Bishop 2006) has recently attracted considerable attention in all scientific disciplines and is highly expected to lead to an in-depth understanding of complex nonlinear combustion dynamics and the development of sophisticated detectors of a precursor of combustion instability (Hachijo et al 2019;Kobayashi et al 2019;Shinchi et al 2021;Waxenegger-Wilfing et al 2021;Guan et al 2022). Gotoda and co-workers (Hachijo et al 2019;Kobayashi et al 2019) attempted to detect a precursor of intermediate-frequency combustion instability in a swirl-stabilized combustor using a combinational methodology of the ordinal partition transition network or Jensen-Shannon complexity-entropy causality plane (Rosso et al 2007), k-means clustering (Lloyd 1982) and a support vector machine (Vapnik 1999).…”

“…The early detection of combustion instability in a staged multisector combustor (Shinchi et al 2021) and a cryogenic rocket thrust chamber (Waxenegger-Wilfing et al 2021) has more recently been examined, both of which consider recurrence-based analysis (Marwan et al 2007) and the support vector machine. Guan et al (2022) elucidated the relationship between the two interactions: pressure-pressure and pressure-heat release rate interactions during intermediate-frequency combustion instability in coupled swirl-stabilized combustors using a combinational methodology of the joint recurrence network (Marwan et al 2007) and Gaussian mixture model clustering (Reynolds 2009).…”

“…We finally introduce a clustering-based transition network using the spectral-clustering method (Von Luxburg 2007) in terms of machine learning. Note that, similarly to the motivation of Guan et al (2022), we employ spectral clustering as unsupervised machine learning to reveal the hidden network dynamics of unlabelled physical quantities (pressure, flow velocity of the fuel jet and temperature) associated with the feedback mechanism of high-frequency combustion instability in our combustor, rather than to detect a precursor of combustion instability using a new data set.…”

“…Guan et al. (2022) elucidated the relationship between the two interactions: pressure–pressure and pressure–heat release rate interactions during intermediate-frequency combustion instability in coupled swirl-stabilized combustors using a combinational methodology of the joint recurrence network (Marwan et al. 2007) and Gaussian mixture model clustering (Reynolds 2009).…”