In this communication, we investigate the dynamics of a simple model of reed musical instrument with a time-varying blowing pressure accounting for attack transients performed by the musician. Indeed, when playing a reed instrument (such as the clarinet), initial acoustical transients (i.e., beginning of a note) are of critical importance. In practice, we study a toy model written as a one-dimensional non-autonomous dynamical system obtained by slowly varying in time a bifurcation parameter of the corresponding autonomous model with constant bifurcation parameter. The study focuses on the case where the time-varying parameter crosses the bistability region of the autonomous system. In the framework of the geometric singular perturbation theory (GSPT), we show that a single solution of the system can be used to describe its the global behavior. This allows to predict, depending on initial conditions and rate of change of the blowing pressure, whether or not a sound is produced during transient.

In this paper, we investigate the dynamics of particles within a bi-quartic potential well, characterized by the coupled potential function \( V(x, y) = \frac{1}{2}x^2 + \frac{1}{2}y^2 - \frac{1}{4}x^4 - \frac{1}{4}y^4 + Cx^2y^2 \). Our focus is on the safe basins of escape (SBoE) and level-crossing (SBoLC) within the initial displacement conditions (IDCs) plane, where the particle's motion is bounded. The study is limited to uniform initial velocity conditions, although extending it to arbitrary initial conditions (ICs) is straightforward. The coupling term allows energy exchange between the modes, enabling IDCs with enough energy to potentially escape the well over time. We find that escape trajectories often pass near one of the four saddles of the potential. Numerical simulations reveal that the SBoE has fractal boundaries because of the energy exchange mechanism. To address safety-critical applications where these chaotic regimes must be avoided, we introduce a factor of safety (FoS) that leads to the definition of a safety region (SR). Crossing the SR's boundary shifts the problem from escape to level-crossing (LC). Assuming harmonic-like solutions of the differential equations, which have slowly varying amplitude and phase values, we transform our system into an appropriate form for averaging. By eliminating time as a variable and realizing that only the phase difference is significant, we derive two first integrals of the particle motion in analytic form, which allows us to analytically determine the SR boundary and calculate its size based on the coupling parameter $C$.

Network dynamics is nowadays of extreme relevance to model and analyze complex systems. From a dynamical systems perspective, understanding the local behavior near equilibria is of utmost importance. In particular, equilibria with at least one zero eigenvalue play a crucial role in bifurcation analysis. In this presentation, we want to shed some light on nilpotent equilibria of network dynamical systems. As a main result, we show that the blow-up technique, which has proven to be extremely useful in understanding degenerate singularities in low-dimensional ordinary differential equations, is also suitable in the framework of network dynamical systems. Most importantly, we show that the blow-up technique preserves the network structure. The further usefulness of the blow-up technique, especially with regard to the desingularization of a nilpotent point, is showcased for an adaptive network of Kuramoto oscillators.

We extend the current theory for spectral submanifolds (SSMs) to ones covering slowly varying dynamical systems. Restricting the dynamics onto these SSMs provides one with a mathematically rigorous model reduction technique. We also observe a justified approximation of adiabtic SSMs can be obtained from data. We demonstrate these capabilities for controlling a high- dimensional nonlinear finite element model describing a soft trunk robot. We formulate a novel model predictive control scheme using adiabatic SSMs constructed from data. We find our method outperforms the state-of-the-art techniques for trajectory tracking control tasks.