MS05.3: Slow-Fast Systems and Phenomena

Acoustic and free-surface gravity waves have been considered as separate phenomena due to the disparate length- and time-scales that characterises them. Recent studies show that acoustic-gravity wave interaction is possible via a nonlinear triad mechanism. Such interaction allows energy exchange which could explain various natural phenomena. Here, attention is focused on the a large-scale nonlinear interaction, between atmospheric acoustic waves and surface ocean waves. As a case study we consider the 2022 Hunga Tonga-Hunga Ha'apai eruption and tsunami, and show that such a nonlinear resonance mechanism can explain the generation and amplification of the tsunami.

Mixed-mode oscillations (MMOs) are characterised by a pattern of small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs). They typically occur in systems with multiple time scales, for which intersections between slow manifolds are known to organise the number of SAOs. We investigate the mechanism underlying their generation in the four-dimensional Olsen model for peroxidase-oxidase reaction. In this four-dimensional phase space, we find that the SAOs come about due to the geometry of a surface of heteroclinic connections between two saddle slow manifolds. We employ an advanced numerical set-up to compute the surface of heteroclinic connections through a region of phase space that exhibits a slow passage through a Hopf bifurcation, where the SAOs are generated. Moreover, we compute its intersection curves with two well-chosen hyperplanes. In this way, we show how the surface of heteroclinic connections delimits rotational sectors with given numbers of subsequent SAOs. We also illustrate its role in determining the exit from this region of SAOs and the onset of LAOs.

The autoresonant approach to excitation of space-time quasicrystals in Bose-Einstein condensates by two chirped frequency parametric drivings is proposed and studied within Gross-Pitaevskii equation. Weakly nonlinear theory of the process is developed using Whitham’s averaged variational principle. The theory also describes the threshold phenomenon setting the lowest bound on the driving amplitudes for autoresonant excitation