A Transfer Operator Approach to Modeling Deterministic and Stochastic Transport, with Applications in the Physical Sciences

Clarkson University, Potsdam NY

Investigators

Abstract

Proposal: DMS-0404778 PI: Erik M Bollt Institution: Clarkson University Title: A Transfer Operator Approach to Modeling Deterministic and Stochastic Transport, with Applications in the Physical Sciences ABSTRACT While highly successful, the stochastic ordinary differential equations (SDEs) and stochastic partial differential equations (SPDEs), have a high overhead of background material to learn. Traditional methods become particularly complex when coping with real problems, which may generally be high dimensional, with general noise profiles, and have nonlinear interactions. Directly approximating the action of the stochastic Frobenius-Perron operator offers a practical and accessible tool. The investigator develops computational methods to model and identify transport activity in both deterministic and stochastically perturbed dynamical systems. The investigator develops an analysis of the transfer operator beyond their usual theoretical use, into a broad and unified suite of computational tools for solving practical problems. The investigator is developing important applications of the computational methods including: 1) mapping the phase space for regions corresponding to high transport flux activity, leading for example to noise-induced bursting in multi-stable systems, 2) developing control of transport algorithms, to either amplify or decrease bursting activity through low-energy control inputs, 3) developing efficient and low-dimensional models of the transfer operator from experimental data, to do system identification and parameter estimation within the global perspective of the transfer operator, 4) developing low-dimensional, nonparametric statistical hypothesis tests to identify nonstationarity and significant system changes, or system "damage," in high dimensional data sets known only through measured data. Even though the complex oscillations seen in the physical world around us have been a subject of intense study in practically every branch of science and engineering, the traditional tools for their analysis remain somewhat technical to learn and apply to practical problems when noise is involved. There are many examples in which the interaction between determinism and a small noise component can give rise to complicated motions that would not occur without the interaction of both mechanisms. Mapping mechanism and timing of noise induced bursting has implications of important social impact. For example, the investigator is studying population dynamics of disease spread, in which external noise excitation can lead to complicated oscillations, and large bursts corresponding to epidemic. Understanding the timing and mechanisms behind these epidemics together with control algorithms leads to suggesting a radically new vaccination protocal in which well timed but relatively noninvasive intervention will result in averting the problem. Similarly, investigations of noise induced transitions of a mechanical beam structure, in nonlinear optics (a noisy lasers), in electronic circuits, in the dynamics of air-pollution, and a reaction diffusion system of a chemical oscillator shows that these developing tools have wide ranging application and importance. Development of damage detection tests of nonstationarity with global perspective suggests a powerful new way to observe when a system has been radically changed, suggesting algorithms for a new class of better sensors to avert hazards.