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Develop higher-order averaging for multiplicative linear-feedback forcing

Develop an averaging method that incorporates higher-order terms for the extended Stuart–Landau oscillator with multiplicative periodic forcing of the form A x cos(Ω t), so that the averaged dynamics capture the amplitude variation observed during 1:1 synchronization and resolve the inconsistency with lowest-order averaging, which eliminates the perturbation.

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Background

For the case of a periodic force combined with linear feedback (A x cos Ω t), the lowest-order averaging used by the authors cancels the perturbation in the averaged system, predicting synchronization only at the natural frequency and no amplitude modulation.

However, numerical simulations show a finite synchronization interval and clear amplitude variation under 1:1 locking. The authors explicitly point out this disagreement and identify the need for an averaging treatment that retains higher-order effects to reconcile analysis with simulations.

References

Several open questions remain to be examined. Second, when considering a periodic force combined with linear feedback [Eq. esl_mp], the perturbation averages out by the lowest-order averaging method. This is inconsistent with numerical results, in which the amplitude of the oscillator varies during 1:1 synchronization [Fig. \ref{reso_esl_multi_freq} (a)]. This disagreement between analysis and numerical simulation would be solved by an averaging method considering the effects of higher-order terms.

— Periodic forces combined with feedback induce quenching in a bistable oscillator (2405.19929 - Kato et al., 30 May 2024) in Section 6 (Discussion), paragraph beginning “Several open questions remain to be examined.”, Second item