Abstract

During the last decades, we have witnessed various micro systems revolutionized fundamental and applied science. Due to their small size and low damping, these devices often exhibit significant nonlinearity and thus the operational range of these impressive applications shrinks. Therefore, understanding the mechanisms leading to nonlinearity in such systems will eliminate obstacles to their further development and significantly enhance their performance. Motivated by the need to advance current capabilities of micro systems, our research has been focused on the implementation of intentional intrinsic nonlinearity in the design of micro resonators and proved that harnessing intentional strong nonlinearity enables exploiting various nonlinear phenomena, not attainable in linear settings, such as broadband resonances, dynamic instabilities, nonlinear hysteresis, and passive targeted energy transfers.

We developed a comprehensive analytical, numerical, and experimental methodology to consider structural nonlinearity as a main design factor enabling to tailor mechanical resonances and achieve targeted performance. We investigated the mechanism of geometric nonlinearity in a non-prismatic microresonator and suggested strategies to tailor the various types of nonlinear resonance. Our more recent works focus on exploiting nonlinearity and multimodality simultaneously by internally coupling two or more modes through the mechanism of internal resonance or combination resonance. This talk will introduce various types of nonlinear phenomena realized in micro systems and discuss their unique behavioral features that can be exploited in the field of Micro-Electro-Mechanical Systems (MEMS) and Atomic Force Microscopy (AFM).

Abstract

A variety of large dimensional systems, ranging from engineering to climate sciences and ecology, are at risk of critical transitions. These systems can shift abruptly from one state to another when parameters that slowly and smoothly drift cross a threshold. It is exceedingly difficult to know if a system comes close to critical transitions because typically there are no easily noticeable changes in the system dynamics until it is too late and the transition has occurred. Furthermore, accurate models of many natural and engineered systems are often not available, and predictions based on incomplete models have limited accuracy. Thus, a significant challenge emerges. How could we forecast such transitions before they occur? The answer lies in a combined use of invariants in nonlinear dynamics and data-driven methods that together can predict such catastrophic events. 

In this talk, we introduce a unique set of data-driven approaches developed to forecast critical points and post-critical dynamics using measurements of the system response collected only in the pre-transition regime. The forecasting approach is based on the phenomenon of critical slowing down, namely the slow dynamics systems exhibit near a tipping point. Based on observations of the system response to natural or controlled perturbations, the method discovers system’s stability, resilience, and equilibria in current and upcoming conditions. The application of this finding in physical experiments and computational methods is demonstrated for a variety of natural and engineered systems including microsensors (vibration-based mass detectors), aeroelastic systems (flutter of 2D airfoils and 3D wings), traffic flow systems (onset of traffic jams), and population dynamical systems (yeast populations, ecological systems). 

Abstract

Melnikov's methods, which enable us to detect the existence of transverse homo- and heteroclinic orbits and periodic orbits and their bifurcations, is now one of classical techniques to study forced nonlinear oscillators. In particular, if an integral called the Melnikov integral or function has a simple zero, then these orbits exist. Recently, similar approaches have been developed to show the nonintegrability of such systems which are not necessarily Hamiltonian. Here non-Hamiltonian systems are generally called integrable if they have a sufficient number of first integrals and commutative vector fields.

In this talk, I present the approaches to prove the nonintegrability of forced nonlinear oscillators and demonstrate them for some examples, based on the following recent papers of my coworker and me:
[1] S. Motonaga and K. Yagasaki, Obstructions to integrability of nearly integrable dynamical systems near regular level sets, submitted for publication.
[2] S. Motonaga and K. Yagasaki, Nonintegrability of forced nonlinear oscillators, submitted.
[3] K. Yagasaki, Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits, J Nonlinear Sci. 32 (2022), 4.
[4] K. Yagasaki, Nonintegrability of the restricted three-body problem, submitted for publication.
[5] K. Yagasaki, Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near the unperturbed homo- and heteroclinic orbits, submitted for publication. 

I begin with rather standard results of Melnikov’s method for forced nonlinear oscillators. I next give a general definition of integrability for non-Hamiltonian systems and explain its meaning. After briefly reviewing the work of Poincare and Kozlov, the differential Galois theory and Morales-Ramis theory, I describe some techniques to prove the nonintegrability of forced nonlinear oscillators and illustrate them for several types of the Duffing oscillators. Finally, I mention further related theoretical results and applications.