Phase sensitivity analysis is a powerful method for studying (asymptotically periodic) bursting neuron models. One popular way of capturing phase sensitivity is through the computation of isochrons---subsets of the state space that each converge to the same trajectory on the limit cycle. However, the computation of isochrons is notoriously difficult, especially for bursting neuron models. In [W. E. Sherwood and J. Guckenheimer, SIAM J. Appl. Dyn. Syst., 9 (2010), pp. 659--703], the phase sensitivity of the bursting Hindmarsh--Rose model is studied through the use of singular perturbation theory: cross sections of the isochrons of the full system are approximated by those of fast subsystems. In this paper, we complement the previous study, providing a detailed phase sensitivity analysis of the full (three-dimensional) system, including computations of the full (two-dimensional) isochrons. To our knowledge, this is the first such computation for a bursting neuron model. This was made possible thanks to the numerical method recently proposed in [A. Mauroy and I. Mezić, Chaos, 22 (2012), 033112]---relying on the spectral properties of the so-called Koopman operator---which is complemented with the use of adaptive quadtree and octree grids. The main result of the paper is to highlight the existence of a region of high phase sensitivity called the almost phaseless set and to completely characterize its geometry. In particular, our study reveals the existence of a subset of the almost phaseless set that is not predicted by singular perturbation theory (i.e., by the isochrons of fast subsystems). We also discuss how the almost phaseless set is related to empirically observed phenomena such as addition/deletion of spikes and to extrema of the phase response of the system. Finally, through the same numerical method, we show that an elliptic bursting model is characterized by a very high phase sensitivity and other remarkable properties.

We present a numerical method for minimum time heading control of fixed speed AUVs (autonomous underwater vehicles) such as gliders in known, spatially complex, 2D, time-varying flow fields. This problem is difficult because locally optimal trajectories abound and, worse, currents stronger than the vehicle can push it far off course. Nevertheless, globally optimal trajectories may be obtained (where they exist) by solving a dynamic HJB (Hamilton Jacobi Bellman) partial differential equation for the time-varying optimal time-to-go function and the associated optimal feedback control law; the local optima and strong currents simply make the control law and the time-to-go function, respectively, discontinuous. In prior work, we found solutions via a variant of the “extremal field” method–essentially the method of characteristics, and equivalent to tracking a 2D “controllability front” backward-in-time from the target set, a line in space-time. In the present work, we exploit a special property of minimum time control to obtain the same globally optimal trajectories, albeit in open loop form, by tracking a 1D “reachability front” forward-in-time from the initial position. The method is further improved by a trimming procedure for locally optimal trajectories. It is tested on a numerically defined flow field from a model of the Adriatic Sea.

We study path planning for small vehicles in strong, spatially complex, time-varying flow fields. Of particular interest is how optimal trajectories relate to flow structures and might be approximated heuristically. Toward this end, we focus on cases where the only concern is the position at some fixed final time, and the control effort. This allows a natural coordinate transformation for the optimal control problem in terms of the so-called flow map. In the transformed coordinates the flow is zero, but the control input (the velocity of the vehicle relative to the flow) is multiplied (and, in more than 1 dimension, rotated) by a time-varying matrix-the Jacobian of the flow map. The definition of what we call the pulled back end cost function provides additional insight and leads to a simple but effective "Lagrangian heuristic control" law, which, in 1d at least, reduces to the optimal control for the case of linear time-invariant flows and quadratic end costs. We demonstrate this control and compare it to the optimal control by solving the associated Hamiltonian Jacobi Bellman (HJB) equation backwards in time with an adaptive 1d grid.

This paper presents a novel tabulation strategy for the adaptive numerical integration of chemical kinetics using the computational singular perturbation (CSP) method. The strategy stores and reuses CSP quantities required to filter out fast dissipative processes, resulting in a non-stiff chemical source term. In particular, non-parametric regression on low-dimensional slow invariant manifolds (SIMs) in the chemical state space is used to approximate the CSP vectors spanning the fast chemical subspace and the associated fast chemical time-scales. The relevant manifold and its dimension varies depending on the local number of exhausted modes at every location in the chemical state space. Multiple manifolds are therefore tabulated, corresponding to different numbers of exhausted modes (dimensions) and associated radical species. Non-parametric representations are inherently adaptive, and rely on efficient approximate-nearest-neighbor queries. As the CSP information is only a function of the non-radical species in the system and has relatively small gradients in the chemical state space, tabulation occurs in a lower-dimensional state space and at a relatively coarse level, thereby improving scalability to larger chemical mechanisms. The approach is demonstrated on the simulation of homogeneous constant pressure H2–air and CH4–air ignition, over a range of initial conditions. For CH4–air, results are shown that outperform direct implicit integration of the stiff chemical kinetics while maintaining good accuracy.

We study the problem of steering a vehicle from its initial position in a 2D, time-varying, ocean flow field to a desired target position in minimum time. In particular, we focus on the case where the magnitude of the flow field sometimes exceeds the speed of the vehicle, and thus controllability is an issue. In order to obtain globally optimal, closed loop trajectories, one solves a dynamic Hamilton Jacobi Bellman equation for the optimal “time-to-go” and associated optimal feedback control law. We do this indirectly via a simple but powerful extremal field algorithm, which allows incremental refinement of the solution and is trivial to parallelize. We characterize solutions and the resulting closed loop optimal trajectories for a time-invariant double gyre flow field and for a numerically-defined, time-varying flow field from a real model of the Adriatic Sea.

First Wafer Delay (FWD) has been identified as a barrier to reduced fab cycle time and increased equipment productivity. As such, initiatives to reduce first wafer delay have been proposed as a focal point for next generation semiconductor fabs, including both 450mm fabs as well as "300mm Prime" factories. FWD and Setup Time are major detractors for cycle time reduction, in particular for small lot manufacturing. The definition of FWD, causes, as well as methods for improving FWD are discussed, in addition to recommendations for future equipment and factory design.