Browsing by Subject "Stochastic differential delay equations"
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Item Open Access Dynamical effects of noise on nonlinear systems(2014) Duman, ÖzerRandomness and nonlinear dynamics consitute the most essential part of many events in nature. Therefore, a better and comprehensive understanding of them is a crucial step in describing natural phenomena as well as the prospect of predicting their future outcome. Besides the interest from a fundamental point of view, it is also useful in a wide variety of applications requiring delicate and careful use of energy. Especially recent advances in micro- and nano-scale technology requires harnessing the underlying noise itself as it is relatively hard to exert forces without damaging the system at that scale. The main aim of this work is to study the effects of noise on nonlinear dynamics. We show that the interplay between noise, nonlinearity and nonequilibrium conditions leads to a finite drift with the potential to change the dynamics of the system completely in a predictable and tunable fashion. We report that the noise-induced drift disrupts the phase space of a 2-D nonlinear system by shifting the fixed point by a finite amount which may result in dramatic alterations over the temporal behavior of the system. We track such alterations to several multi-dimensional model systems from ecology, soft matter and statistical physics. In a 2-D ecological model describing two species competing for the same resource, it is found that the system switches between coexistence and extinction states depending on the shift due to the noise-induced drift whereas for an aggregate of Brownian particles, it is shown that noise-induced drift selectively shifts the probability distribution in certain geometries which can be used in the realization of a microparticle sorter in the mould of Feynman ratchets. In the case of the aggregate consisting of microswimmers, tunable anomalous diffusion depending on the confinement length is reported.Item Open Access An SDE approximation for stochastic differential delay equations with state-dependent colored noise(Polymat Publishing, 2016-11) McDaniel, A.; Duman, Ö.; Volpe, G.; Wehr, J.We consider a general multidimensional stochastic differential delay equation (SDDE) with state-dependent colored noises. We approximate it by a stochastic differential equation (SDE) system and calculate its limit as the time delays and the correlation times of the noises go to zero. The main result is proven using a theorem about convergence of stochastic integrals by Kurtz and Protter. It formalizes and extends a result that has been obtained in the analysis of a noisy electrical circuit with delayed state-dependent noise, and may be used as a working SDE approximation of an SDDE modeling a real system where noises are correlated in time and whose response to noise sources depends on the system's state at a previous time.