Forced synchronization of self-excited chaotic thermoacoustic oscillations

Abstract

We experimentally investigate the forced synchronization of a self-excited chaotic thermoacoustic oscillator with two natural frequencies, $f_1$ and $f_2$ . On increasing the forcing amplitude, $\epsilon _f$ , at a fixed forcing frequency, $f_f$ , we find two different types of synchronization: (i) $f_f/f_1 = 1:1$ or $2:1$ chaos-destroying synchronization (CDS), and (ii) phase synchronization of chaos (PSC). En route to $1:1$ CDS, the system transitions from an unforced chaotic state ( ${\rm {CH}}_{1,2}$ ) to a forced chaotic state ( ${\rm {CH}}_{1,2,f}$ ), then to a two-frequency quasiperiodic state where chaos is destroyed ( $\mathbb {T}^2_{2,f}$ ), and finally to a phase-locked period-1 state ( ${\rm {P1}}_f$ ). The route to $2:1$ CDS is similar, but the quasiperiodic state hosts a doubled torus $(2\mathbb {T}^2_{2,f})$ that transforms into a phase-locked period-2 orbit $({\rm {P2}}_f)$ when CDS occurs. En route to PSC, the system transitions to a forced chaotic state ( ${\rm {CH}}_{1,2,f}$ ) followed by a phase-locked chaotic state, where $f_1$ , $f_2$ and $f_f$ still coexist but their phase difference remains bounded. We find that the maximum reduction in thermoacoustic amplitude occurs near the onset of CDS, and that the critical $\epsilon _f$ required for the onset of CDS does not vary significantly with $f_f$ . We then use two unidirectionally coupled Anishchenko–Astakhov oscillators to phenomenologically model the experimental synchronization dynamics, including (i) the route to $1:1$ CDS, (ii) various phase dynamics, such as phase drifting, slipping and locking, and (iii) the thermoacoustic amplitude variations in the $f_f/f_1$ – $\epsilon _f$ plane. This study extends the applicability of open-loop control further to a chaotic thermoacoustic system, demonstrating (i) the feasibility of using an existing actuation strategy to weaken aperiodic thermoacoustic oscillations, and (ii) the possibility of developing new active suppression strategies based on both established and emerging methods of chaos control.