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We present stochastic consensus and convergence of the discrete consensus-based optimization (CBO) algorithm with random batch interactions and heterogeneous external noises. Despite the wide applications and successful performance in many practical simulations, the convergence of the discrete CBO algorithm was not rigorously investigated in such a generality. In this work, we introduce a generalized discrete CBO algorithm with a weighted representative point and random batch interactions, and show that the proposed discrete CBO algorithm exhibits stochastic consensus and convergence toward the common equilibrium state exponentially fast under suitable assumptions on system parameters. For this, we recast the given CBO algorithm with random batch interactions as a discrete consensus model with a random switching network topology, and then we use the mixing property of interactions over sufficiently long time interval to derive stochastic consensus and convergence estimates in mean square and almost sure senses. Our proposed analysis significantly improves earlier works on the convergence analysis of CBO models with full batch interactions and homogeneous external noises.

Microbioreactors operated in real environments are often subject to noise from the environment. This is commonly manifested as fluctuations in the flow rates of the feed streams. Previous studies with larger bioreactors have shown that noise can seriously impair the performance. Given this possibility, the effects of noise on the performance of a microbioreactor have been analyzed for the trans-esterification of vinyl butyrate by 1-butanol by immobilized lipase B to produce butyl butyrate. As in previous work for macrobioreactors, the analysis was done with (i) no noise, (ii) unfiltered noise, and (iii) noise filtered by four different methods, and the fractal dimension of the product was used as an index of the performance.

All fractal dimensions decreased with increasing dilution rates, and significant stochastic chaos was likely at low dilution rates. Of the four types of filters, the auto-associative neural filter (ANF) was the most effective in reducing chaos and restoring of smooth, nearly noise-free performance. The ANF also does not require a process model, which is a significant advantage for real systems. Simulations also revealed that even in the absence of noise, deterministic chaos is possible at low dilution rates; this underscores the importance of efficient filtering under such conditions when external noise too is present. The results thus establish the importance of noise in microbioreactor behavior and the usefulness of the fractal dimension in characterizing the effects.

Weak electrical noise applied in the water around small paddlefish, Polyodon spathula, increases the spatial range over which they can detect and capture planktonic prey (Daphnia), demonstrating stochastic resonance at the level of an animal's feeding behavior. Here we show that optimal-amplitude (~ 0.5 μ V·cm^-1) noise causes a fish to prefer more vertical angles of attack when striking at prey, as revealed in polar graphs. Increased spatial range is also seen in horizontal directions, as outlying shoulders in the probability distribution of horizontal strike distances. High levels of noise increased the distance that approaching prey travelled along the rostrum (an elongated appendage anterior to the head, functioning as an electrosensitive antenna), before the fish first showed a visible fin or body motion in response. There was no significant effect of optimal-amplitude noise on the rate of strikes, although high-amplitude noise reduced the strike rate. The behavioral data were confirmed in neurophysiological experiments demonstrating that stochastic resonance occurs in individual electroreceptors, and in fact occurs at a similar optimal noise level as in behavioral experiments. We conclude that stochastic resonance can be demonstrated in the behavior of animals, and that animals can make use of the increased sensory information available during near-threshold environmental noise.

We investigated, analytically and by analog simulation, a simple model system, an overdamped Kramers oscillator with two additive noise sources, internal white and external colored. We found that the magnetude of noise background in the spectrum of system outpit can decrease when the amplitude of external noise or periodic signal increases.

We review the effect of spatiotemporal noise, white in time and colored in space, on front propagation in systems of reacting and dispersing particles, where the particle motion displays inertia or persistence. We discuss the three main approaches that have been developed to describe transport with inertia, namely hyperbolic reaction-diffusion equations, reaction-Cattaneo systems or reaction-telegraph equations, and reaction random walks. We focus on the mean speed of Fisher waves in these systems and study in particular reaction random walks, which are the most natural generalization of reaction-diffusion equations. Hyperbolic reaction-diffusion equations account for inertia in the transport process in an ad hoc way, whereas the other reaction-transport systems have a proper macroscopic or microscopic foundation. For the former, external noise affects neither the mean wave speed nor the region in parameter space for which Fisher waves exist. For the latter, external noise increases the mean wave speed of Fisher waves and decreases the upper limit for the characteristic time of the transport process, below which propagating fronts exist.

After recalling key phenomenological properties of glass formation, we point out that similar features are exhibited by the dynamical properties of the noise-perturbed iterates of the logistic map at the onset of chaos. The analysis includes two-step relaxation, aging, subdiffusion and arrest, as well as an expression analogous to the Adam–Gibbs relation connecting dynamical and thermodynamic properties of a glass former. The dynamical properties of the logistic map in the presence of external noise are seen to be comparable to those of a supercooled liquid above a glass transition temperature, whereas the noiseless attractor displays typical nonequilibrium aspects like loss of time translation invariance (aging). Reference is made to connections between the noiseless dynamics at the chaos threshold and the nonextensive formalism.

This paper establishes novel criteria for the induced $l_{\infty }$ stability to avoid overflow oscillations in fixed-point digital filters with generalized overflow non-linearities and external noise. The proposed linear matrix inequality (LMI)-based criteria ensure exponential stability as well as confirm reduction in the influence of external noise. The generalized overflow non-linearities which are considered for analysis commonly occur in practice, viz. saturation, zeroing, two's complement, and triangular. The presented approach unifies a string of existing results which are derived by considering saturation non-linearities and external interference. Simulation examples are shown to validate the usefulness of the proposed approach.

Weak electrical noise applied in the water around small paddlefish, Polyodon spathula, increases the spatial range over which they can detect and capture planktonic prey (Daphnia), demonstrating stochastic resonance at the level of an animal’s feeding behavior. Here we show that optimal-amplitude (∼0.5 μV·cm⁻¹) noise causes a fish to prefer more vertical angles of attack when striking at prey, as revealed in polar graphs. Increased spatial range is also seen in horizontal directions, as outlying shoulders in the probability distribution of horizontal strike distances. High levels of noise increased the distance that approaching prey travelled along the rostrum (an elongated appendage anterior to the head, functioning as an electrosensitive antenna), before the fish first showed a visible fin or body motion in response. There was no significant effect of optimal-amplitude noise on the rate of strikes, although high-amplitude noise reduced the strike rate. The behavioral data were confirmed in neurophysiological experiments demonstrating that stochastic resonance occurs in individual electroreceptors, and in fact occurs at a similar optimal noise level as in behavioral experiments. We conclude that stochastic resonance can be demonstrated in the behavior of animals, and that animals can make use of the increased sensory information available during near-threshold environmental noise.

First, we analyze trajectories inside the Feigenbaum attractor and obtain the atypical weak sensitivity to initial conditions and loss of information associated to their dynamics. We identify the Mori singularities in its Lyapunov spectrum with the appearance of a special value for the entropic index q of the Tsallis statistics. Secondly, the dynamics of iterates at the noise-perturbed transition to chaos is shown to exhibit the characteristic elements of the glass transition, e.g. two-step relaxation, aging, subdiffusion and arrest. The properties of the bifurcation gap induced by the noise are seen to be comparable to those of a supercooled liquid above a glass transition temperature.