14 research outputs found
Most probable transition paths in piecewise-smooth stochastic differential equations
We develop a path integral framework for determining most probable paths in a
class of systems of stochastic differential equations with piecewise-smooth
drift and additive noise. This approach extends the Freidlin-Wentzell theory of
large deviations to cases where the system is piecewise-smooth and may be
non-autonomous. In particular, we consider an dimensional system with a
switching manifold in the drift that forms an dimensional hyperplane
and investigate noise-induced transitions between metastable states on either
side of the switching manifold. To do this, we mollify the drift and use
convergence to derive an appropriate rate functional for the system in
the piecewise-smooth limit. The resulting functional consists of the standard
Freidlin-Wentzell rate functional, with an additional contribution due to times
when the most probable path slides in a crossing region of the switching
manifold. We explore implications of the derived functional through two case
studies, which exhibit notable phenomena such as non-unique most probable paths
and noise-induced sliding in a crossing region.Comment: 38 pages, 9 figure
Nature's forms are frilly, flexible, and functional
A ubiquitous motif in nature is the self-similar hierarchical buckling of a
thin lamina near its margins. This is seen in leaves, flowers, fungi, corals
and marine invertebrates. We investigate this morphology from the perspective
of non-Euclidean plate theory. We identify a novel type of defect, a
branch-point of the normal map, that allows for the generation of such complex
wrinkling patterns in thin elastic hyperbolic surfaces, even in the absence of
stretching. We argue that branch points are the natural defects in hyperbolic
sheets, they carry a topological charge which gives them a degree of
robustness, and they can influence the overall morphology of a hyperbolic
surface without concentrating elastic energy. We develop a theory for branch
points and investigate their role in determining the mechanical response of
hyperbolic sheets to weak external forces. We also develop a discrete
differential geometric (DDG) framework for applications to the continuum
mechanics of hyperbolic elastic sheets.Comment: 35 pages, 26 figure
Tipping in a Low-Dimensional Model of a Tropical Cyclone
A presumed impact of global climate change is the increase in frequency and
intensity of tropical cyclones. Due to the possible destruction that occurs
when tropical cyclones make landfall, understanding their formation should be
of mass interest. In 2017, Kerry Emanuel modeled tropical cyclone formation by
developing a low-dimensional dynamical system which couples tangential wind
speed of the eye-wall with the inner-core moisture. For physically relevant
parameters, this dynamical system always contains three fixed points: a stable
fixed point at the origin corresponding to a non-storm state, an additional
asymptotically stable fixed point corresponding to a stable storm state, and a
saddle corresponding to an unstable storm state. The goal of this work is to
provide insight into the underlying mechanisms that govern the formation and
suppression of tropical cyclones through both analytical arguments and
numerical experiments. We present a case study of both rate and noise-induced
tipping between the stable states, relating to the destabilization or formation
of a tropical cyclone. While the stochastic system exhibits transitions both to
and from the non-storm state, noise-induced tipping is more likely to form a
storm, whereas rate-induced tipping is more likely to be the way a storm is
destabilized, and in fact, rate-induced tipping can never lead to the formation
of a storm when acting alone. For rate-induced tipping acting as a destabilizer
of the storm, a striking result is that both wind shear and maximal potential
velocity have to increase, at a substantial rate, in order to effect tipping
away from the active hurricane state. For storm formation through noise-induced
tipping, we identify a specific direction along which the non-storm state is
most likely to get activated
Shape selection in non-Euclidean plates
We investigate isometric immersions of disks with constant negative curvature
into , and the minimizers for the bending energy, i.e. the
norm of the principal curvatures over the class of isometric
immersions. We show the existence of smooth immersions of arbitrarily large
geodesic balls in into . In elucidating the
connection between these immersions and the non-existence/singularity results
of Hilbert and Amsler, we obtain a lower bound for the norm of the
principal curvatures for such smooth isometric immersions. We also construct
piecewise smooth isometric immersions that have a periodic profile, are
globally , and have a lower bending energy than their smooth
counterparts. The number of periods in these configurations is set by the
condition that the principal curvatures of the surface remain finite and grows
approximately exponentially with the radius of the disc. We discuss the
implications of our results on recent experiments on the mechanics of
non-Euclidean plates
A retinal code for motion along the gravitational and body axes
Self-motion triggers complementary visual and vestibular reflexes supporting image-stabilization and balance. Translation through space produces one global pattern of retinal image motion (optic flow), rotation another. We examined the direction preferences of direction-sensitive ganglion cells (DSGCs) in flattened mouse retinas in vitro. Here we show that for each subtype of DSGC, direction preference varies topographically so as to align with specific translatory optic flow fields, creating a neural ensemble tuned for a specific direction of motion through space. Four cardinal translatory directions are represented, aligned with two axes of high adaptive relevance: the body and gravitational axes. One subtype maximizes its output when the mouse advances, others when it retreats, rises or falls. Two classes of DSGCs, namely, ON-DSGCs and ON-OFF-DSGCs, share the same spatial geometry but weight the four channels differently. Each subtype ensemble is also tuned for rotation. The relative activation of DSGC channels uniquely encodes every translation and rotation. Although retinal and vestibular systems both encode translatory and rotatory self-motion, their coordinate systems differ