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Abstract: Nonlinear oscillators have a broad range of applications in engineering, including rotors, energy harvesters, sensors, and precision timing devices. The dynamics of a single oscillator with cubic nonlinearity and external periodic forcing is surprisingly rich. Depending on parameters, it may admit multiple attractors that may be periodic or chaotic. Their basin boundaries may be fractal. Linking oscillators into arrays and adding noise further complicate their dynamics. I will discuss a method for finding the most probable escape paths from the basins of attractors of noisy oscillators, sensor design, and a few open mathematical problems related to nonlinear oscillators.

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Abstract: Professor Gabriel’s research group develops models, theory, and algorithms for solving problems that arise in infrastructure planning such as: energy, water, transport. These models are typified by a set of autonomous agents (i.e., energy market participants, vehicles) that share a common network. The equilibrium aspects arise since each of the players or subsets of the players compete non-cooperatively with each other for the infrastructure network’s resources. The concatenation of all these optimization problems as well as any system-level constraints results in what is known as an equilibrium problem; typically called a mixed complementarity problem (MCP) or a variation inequality (VI). Such problems generalize the Karush-Kuhn-Tucker (KKT) conditions of nonlinear programs, Nash-Cournot games, as well as many other problems in operations research, engineering and economic systems. These equilibrium problems can also be single-level, wherein all the agents are at the same level or such problems can be multi-level. In the latter case, some famous paradigms include: bilevel optimization (e.g., Stackelberg leader-follower games), attacker-defender interdiction problems and trilevel optimization. Please see Professor Gabriel’s website for further details: http://www.stevenagabriel.umd.edu/ or email him directly at with any questions you might have.

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Abstract: This seminar will provide an overview of how continuous stochastic processes have been applied to the study of animal movement ecology using data from GPS tracking devices. I will present the mathematical foundations of these applications and discuss how we statistically fit the stochastic process models to diverse biological datasets. I will then give an overview of the wide range of applications that my colleagues and I have found for these approaches, including such biological topics as:
1) animal home ranges, migration, and space use
2) behavioral evidence for learning and disease states
3) route-based movement by carnivores
4) consumer-resource interactions

Movement data from GPS tracking devices typically feature a high degree of temporal autocorrelation, often at multiple scales. Over the years, our work has dealt with such data in a variety of statistical contexts, including:
1) timeseries analysis
2) kernel density estimation
3) path estimation via kriging
4) estimation of probability ridges
5) comparative (i.e., phylogenetically controlled) analyses
The talk will present results from joint work with mathematicians Leonid Koralov and Mark Lewis; past-postdocs Christen Fleming, Eliezer Gurarie, and Michael Noonan; past-PhD students Justin Calabrese and Nicole Barbour; current PhD students Frank McBride, Marron McConnell, Gayatri Anand, Stephanie Chia, Qianru Liao, and Phillip Koshute; current undergraduate Zachary Tomares; and hundreds of biologists. Open questions abound and span a wide range of difficulty. I have access to mountains of animal movement data and am eager for collaborators.

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Abstract: This talk shows the use of parameter continuation techniques to characterize intermediate-term dynamics due to the presence of small Brownian noise near normally-hyperbolic, transversally stable periodic orbits and quasiperiodic invariant tori found in the deterministic limit. The proposed formulation relies on adjoint boundary-value problems for constructing continuous families of transversal hyperplanes that are invariant under the linearized deterministic flow, and covariance boundary-value problems for describing Gaussian distributions of intersections of stochastic trajectories with these hyperplanes. Analytical and numerical results, including validation with the help of the continuation package COCO, show excellent agreement with stochastic time integration for problems with either autonomous or time-periodic drift terms.

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Abstract: TBD