Auto-parametric systems consist of at least one oscillator that is directly excited and coupled nonlinearly to a subsystem in such a way that the oscillator can vibrate while the subsystem remains stationary. Small amplitude vibration of the single oscillator or of one subsystem may destabilize the entire system. This destabilization is auto-parametric resonance. The purpose of this work is to examine the stationary motion and stability properties of stationary motion of noisy auto-parametric systems. The nonlinear dynamics of such noisy systems are extremely rich and largely unexplored. Auto-parametric resonance has many important engineering applications. For example, in aircraft, vibration of the "suspenders" connecting the engines and the wings occurs due to turbulent air forcing on the wings -- this is stochastic auto-parametric "resonance". Another example is the heave-roll motion of sea vessels produced by agitated waters; in extreme cases the danger of capsizing arises. Shock absorbers are also susceptible to auto-parametric resonance. While the focus of the present study is on the applications above the results of this research will be applicable even to biological problems such as the mathematical modeling of a pool of spiking neurons in the locus coeruleus, a brain nucleus involved in modulating cognitive performance.
The approach will be cross-disciplinary, spanning the spectrum from analytical techniques and modeling to numerical and experimental verification. The first challenge is to develop an effective, systematic approach to determine the almost-sure stability of single mode nonlinear solutions. In the presence of a separation of scales, where the noise is asymptotically small, one exploits symmetries to use recent mathematical results concerning stochastic averaging to find a lower-dimensional description of the system. Stochastic numerical simulations will be used to verify the bifurcation behavior obtained through analysis and to provide clues to new phenomena which are global in nature and not detected through the theoretical analysis. The PI will recruit students from underrepresented minority and women groups. The project will directly train both graduate and undergraduate students. The broader impacts of the proposal include a partnership with the Structural Department of the University of L'Aquila, Italy, where experimental validations will be made. Students will work closely with professors from the University of L'Aquila's Nonlinear Dynamics Laboratory. The theoretical component of the proposed research will be tightly coupled to the computational and experimental efforts.
