Direct event location techniques based on Adams multistep methods for discontinuous ODEs

In this paper we consider numerical techniques to locate the event points of the differential system x′=f(x), where f is a discontinuous vector field along an event surface splitting the state space into two different regions R1 and R2 and f(x)=fi(x) when x∈Ri, for i=1,2 while f1(x)≠f2(x) when x∈Σ. Methods based on Adams multistep schemes which approach the event surface Σ from one side only and in a finite number of steps are proposed. Particularly, these techniques do not require the evaluation of the vector field f1 (respectively, f2) in the region R2 (respectively R1) and are based on the computation–at each step– of a new time ste

Sliding at first order: Higher-order momentum distributions for discontinuous image registration

In this paper, we propose a new approach to deformable image registration that captures sliding motions. The large deformation diffeomorphic metric mapping (LDDMM) registration method faces challenges in representing sliding motion since it per construction generates smooth warps. To address this issue, we extend LDDMM by incorporating both zeroth- and first-order momenta with a non-differentiable kernel. This allows to represent both discontinuous deformation at switching boundaries and diffeomorphic deformation in homogeneous regions. We provide a mathematical analysis of the proposed deformation model from the viewpoint of discontinuous systems. To evaluate our approach, we conduct experiments on both artificial images and the publicly available DIR-Lab 4DCT dataset. Results show the effectiveness of our approach in capturing plausible sliding motion

Bifurcation and chaos in simple discontinuous systems separated by a hypersurface

This research focuses on a mathematical examination of a path to sliding period doubling and chaotic behaviour for a novel limited discontinuous systems of dimension three separated by a nonlinear hypersurface. The switching system is composed of dissipative subsystems, one of which is a linear systems, and the other is not linked with equilibria. The non-linear sliding surface is designed to improve transient response for these subsystems. A Poincaré return map is created that accounts for the existence of the hypersurface, completely describing each individual sliding period-doubling orbits that route to the sliding chaotic attractor. Through a rigorous analysis, we show that the presence of a nonlinear sliding surface and a set of such hidden trajectories leads to novel bifurcation scenarios. The proposed system exhibits period-m orbits as well as chaos, including partially hidden and sliding trajectories. The results are numerically verified through path-following techniques for discontinuous dynamical systems

Data-Driven Methods to Build Robust Legged Robots

For robots to ever achieve signicant autonomy, they need to be able to mitigate performance loss due to uncertainty, typically from a novel environment or morphological variation of their bodies. Legged robots, with their complex dynamics, are particularly challenging to control with principled theory. Hybrid events, uncertainty, and high dimension are all confounding factors for direct analysis of models. On the other hand, direct data-driven methods have proven to be equally dicult to employ. The high dimension and mechanical complexity of legged robots have proven challenging for hardware-in-the-loop strategies to exploit without signicant eort by human operators. We advocate that we can exploit both perspectives by capitalizing on qualitative features of mathematical models applicable to legged robots, and use that knowledge to strongly inform data-driven methods. We show that the existence of these simple structures can greatly facilitate robust design of legged robots from a data-driven perspective. We begin by demonstrating that the factorial complexity of hybrid models can be elegantly resolved with computationally tractable algorithms, and establish that a novel form of distributed control is predicted. We then continue by demonstrating that a relaxed version of the famous templates and anchors hypothesis can be used to encode performance objectives in a highly redundant way, allowing robots that have suffered damage to autonomously compensate. We conclude with a deadbeat stabilization result that is quite general, and can be determined without equations of motion.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155053/1/gcouncil_1.pd

On the interaction of gamma-rhythmic neuronal populations

Local gamma-band (~30-100Hz) oscillations in the brain, produced by feedback inhibition on a characteristic timescale, appear in multiple areas of the brain and are associated with a wide range of cognitive functions. Some regions producing gamma also receive gamma-rhythmic input, and the interaction and coordination of these rhythms has been hypothesized to serve various functional roles. This thesis consists of three stand-alone chapters, each of which considers the response of a gamma-rhythmic neuronal circuit to input in an analytical framework. In the first, we demonstrate that several related models of a gamma-generating circuit under periodic forcing are asymptotically drawn onto an attracting invariant torus due to the convergence of inhibition trajectories at spikes and the convergence of voltage trajectories during sustained inhibition, and therefore display a restricted range of dynamics. In the second, we show that a model of a gamma-generating circuit under forcing by square pulses cannot maintain multiple stably phase-locked solutions. In the third, we show that a separation of time scales of membrane potential dynamics and synaptic decay causes the gamma model to phase align its spiking such that periodic forcing pulses arrive under minimal inhibition. When two of these models are mutually coupled, the same effect causes excitatory pulses from the faster oscillator to arrive at the slower under minimal inhibition, while pulses from the slower to the faster arrive under maximal inhibition. We also show that such a time scale separation allows the model to respond sensitively to input pulse coherence to an extent that is not possible for a simple one-dimensional oscillator. We draw on a wide range of mathematical tools and structures including return maps, saltation matrices, contraction methods, phase response formalism, and singular perturbation theory in order to show that the neuronal mechanism of gamma oscillations is uniquely suited to reliably phase lock across brain regions and facilitate the selective transmission of information

Fundamental matrix solutions of piecewise smooth differential systems

We consider the fundamental matrix solution associated to piecewise smooth differential systems of Filippov type, in which the vector field varies discontinuously as solution trajectories reach one or more surfaces. We review the cases of transversal intersection and of sliding motion on one surface. We also consider the case when sliding motion takes place on the intersection of two or more surfaces. Numerical results are also given