10 research outputs found
A primer on noise-induced transitions in applied dynamical systems
Noise plays a fundamental role in a wide variety of physical and biological
dynamical systems. It can arise from an external forcing or due to random
dynamics internal to the system. It is well established that even weak noise
can result in large behavioral changes such as transitions between or escapes
from quasi-stable states. These transitions can correspond to critical events
such as failures or extinctions that make them essential phenomena to
understand and quantify, despite the fact that their occurrence is rare. This
article will provide an overview of the theory underlying the dynamics of rare
events for stochastic models along with some example applications
Dynamics of Coupled Particles in a Time-Dependent, Double-Gyre Flow
We consider a time-dependent, wind-driven, stochastic double-gyre flow, and investigate the interaction between the flow and coupled particles operating within the flow. It is known that noise can cause individual particles to escape from one gyre to another gyre. By computing the Lagrangian coherent structures (LCS) of the system, one can determine low and high probability regions of particle escape. We adjust the coupling between two particles, and study the effect on particle escape for a variety of initial conditions and noise intensities
Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning
The mean exit time escaping basin of attraction in the presence of white
noise is of practical importance in various scientific fields. In this work, we
propose a strategy to control mean exit time of general stochastic dynamical
systems to achieve a desired value based on the quasipotential concept and
machine learning. Specifically, we develop a neural network architecture to
compute the global quasipotential function. Then we design a systematic
iterated numerical algorithm to calculate the controller for a given mean exit
time. Moreover, we identify the most probable path between metastable
attractors with help of the effective Hamilton-Jacobi scheme and the trained
neural network. Numerical experiments demonstrate that our control strategy is
effective and sufficiently accurate
Extinction of Species Due to Deterministic and Stochastic Interactions in Food Webs
Previous research on the extinctions that occur in niche model food webs with deterministic and stochastic dynamics has shown that the structure of the food web can play an important role in extinction cascades. In this thesis, other types of synthetic food web models are considered, namely the cascade and generalized cascade models, and the extinction cascades of these food webs are compared with previous findings on the extinction cascades from the niche model. It was found that there are many similarities in the results for all three models, which prompted a closer analysis using food webs with deterministic dynamics. We developed a method to theoretically predict the survival or extinction of species in two- and three-species food webs, and compared the predictions with numerical results
Control of Secondary Extinctions in Stochastic Food Webs
Studies on both model-based and empirical food webs have shown that per- turbations to an ecological community can cause a species to go extinct, often resulting in the loss of additional species in a cascade of secondary extinctions. These eects can seriously debilitate a food web and threaten the existence of an ecosystem. Here, we consider niche model-based food webs with internal noise and investigate the eects of a control on a secondary extinction cas- cade triggered by a noise-induced extinction. We show that the forced removal of a nonbasal species immediately after a primary extinction can extend the mean time to extinction of individual nonbasal species as well as that of the complete extinction cascade. An analysis of numerical and statistical results illustrates the eectiveness of a control in delaying the mean time to extinction for endangered species in stochastic food webs
Learning noise-induced transitions by multi-scaling reservoir computing
Noise is usually regarded as adversarial to extract the effective dynamics
from time series, such that the conventional data-driven approaches usually aim
at learning the dynamics by mitigating the noisy effect. However, noise can
have a functional role of driving transitions between stable states underlying
many natural and engineered stochastic dynamics. To capture such stochastic
transitions from data, we find that leveraging a machine learning model,
reservoir computing as a type of recurrent neural network, can learn
noise-induced transitions. We develop a concise training protocol for tuning
hyperparameters, with a focus on a pivotal hyperparameter controlling the time
scale of the reservoir dynamics. The trained model generates accurate
statistics of transition time and the number of transitions. The approach is
applicable to a wide class of systems, including a bistable system under a
double-well potential, with either white noise or colored noise. It is also
aware of the asymmetry of the double-well potential, the rotational dynamics
caused by non-detailed balance, and transitions in multi-stable systems. For
the experimental data of protein folding, it learns the transition time between
folded states, providing a possibility of predicting transition statistics from
a small dataset. The results demonstrate the capability of machine-learning
methods in capturing noise-induced phenomena
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