Title: Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium
We consider the exit problem for a one-dimensional system with random switching near an unstable equilibrium point of the averaged drift. In the infinite switching rate limit, we show that the exit time satisfies a limit theorem with a logarithmic deterministic term and a random correction converging in distribution. Thus, this setting is in the universality class of the unstable equilibrium exit under small white-noise perturbations. more »« less
Bakhtin, Yuri; Pajor-Gyulai, Zsolt
(, Journal of Applied Probability)
null
(Ed.)
Abstract For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits. We also discuss our program on rare transitions in noisy heteroclinic networks.
Morrison, Megan; Young, Lai-Sang
(, Chaos: An Interdisciplinary Journal of Nonlinear Science)
Key features of biological activity can often be captured by transitions between a finite number of semi-stable states that correspond to behaviors or decisions. We present here a broad class of dynamical systems that are ideal for modeling such activity. The models we propose are chaotic heteroclinic networks with nontrivial intersections of stable and unstable manifolds. Due to the sensitive dependence on initial conditions, transitions between states are seemingly random. Dwell times, exit distributions, and other transition statistics can be built into the model through geometric design and can be controlled by tunable parameters. To test our model’s ability to simulate realistic biological phenomena, we turned to one of the most studied organisms, C. elegans, well known for its limited behavioral states. We reconstructed experimental data from two laboratories, demonstrating the model’s ability to quantitatively reproduce dwell times and transition statistics under a variety of conditions. Stochastic switching between dominant states in complex dynamical systems has been extensively studied and is often modeled as Markov chains. As an alternative, we propose here a new paradigm, namely, chaotic heteroclinic networks generated by deterministic rules (without the necessity for noise). Chaotic heteroclinic networks can be used to model systems with arbitrary architecture and size without a commensurate increase in phase dimension. They are highly flexible and able to capture a wide range of transition characteristics that can be adjusted through control parameters.
Cetemen, Doruk; Urgun, Can; Yariv, Leeat
(, Journal of Political Economy)
We introduce a framework for studying collective search by teams. Discoveries are correlated over time and governed by a Brownian path, where search speed is jointly controlled. Agents individually choose when to cease search and implement their best discovery. We characterize equilibrium and optimal policies. Search speeds are constant within active alliances and depend on complementarities between members. A drawdown stopping boundary governs each agent’s search termination. The consequent exit waves, whereby possibly heterogeneous agents cease search simultaneously, exhibit deterministic sequencing but stochastic timing. We highlight environments with lower than optimal equilibrium speeds and search durations, and different exit waves.
Probabilistic spin logic (PSL) has recently been proposed as a novel computing paradigm that leverages random thermal fluctuations of interacting bodies in a system rather than deterministic switching of binary bits. A PSL circuit is an interconnected network of thermally unstable units called probabilistic bits (p-bits), whose output randomly fluctuates between bits 0 and 1. While the fluctuations generated by p-bits are thermally driven, and therefore, inherently stochastic, the output probability is tunable with an external source. Therefore, information is encoded through probabilities of various configuration of states in the network. Recent studies have shown that these systems can efficiently solve various types of combinatorial optimization problems and Bayesian inference problems that modern computers are unfit for. Previous experimental studies have demonstrated that a single magnetic tunnel junctions (MTJ) designed to be thermally unstable can operate tunable random number generator making it an ideal hardware solution for p-bits. Most proposals for designing an MTJ to operate as a p-bit involve patterning the MTJ as a circular nano-pillar to make the device thermally unstable and then use spin transfer torque (STT) as a tuning mechanism. However, the practical realization of such devices is very challenging since the fluctuation rate of these devices are very sensitive to any device variations or defects caused during fabrication. Despite this challenge, MTJs are still the most promising hardware solution for p-bits because MTJs are very unique in that they can be tuned by multiple other mechanisms such spin orbit torque, magneto-electric coupling, and voltage-controlled exchange coupling. Furthermore, multiple forces can be used simultaneously to drive stochastic switching signals in MTJs. This means there are a large number of methods to tune, or termed as bias, MTJs that can be implemented in p-bit circuits that can alleviate the current challenges of conventional STT driven p-bits. This article serves as a review of all of the different methods that have been proposed to drive random fluctuations in MTJs to operate as a probabilistic bit. Not only will we review the single-biasing mechanisms, but we will also review all the proposed dual-biasing methods, where two independent mechanisms are employed simultaneously. These dual-biasing methods have been shown to have certain advantages such as alleviating the negative effects of device variations and some biasing combinations have a unique capability called ‘two-degrees of tunability’, which increases the information capacity in the signals generated.
Bakhtin, Yuri, and Gaál, Alexisz. Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium. Retrieved from https://par.nsf.gov/biblio/10230697. Stochastics and Dynamics 20.04 Web. doi:10.1142/S0219493720500264.
Bakhtin, Yuri, & Gaál, Alexisz. Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium. Stochastics and Dynamics, 20 (04). Retrieved from https://par.nsf.gov/biblio/10230697. https://doi.org/10.1142/S0219493720500264
@article{osti_10230697,
place = {Country unknown/Code not available},
title = {Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium},
url = {https://par.nsf.gov/biblio/10230697},
DOI = {10.1142/S0219493720500264},
abstractNote = {We consider the exit problem for a one-dimensional system with random switching near an unstable equilibrium point of the averaged drift. In the infinite switching rate limit, we show that the exit time satisfies a limit theorem with a logarithmic deterministic term and a random correction converging in distribution. Thus, this setting is in the universality class of the unstable equilibrium exit under small white-noise perturbations.},
journal = {Stochastics and Dynamics},
volume = {20},
number = {04},
author = {Bakhtin, Yuri and Gaál, Alexisz},
editor = {null}
}
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