Denis S Goldobin

Abstract: We consider application of the multiple time delayed feedback for control of anharmonic (nonlinear) oscillators subject to noise. In contrast to the case of a single delay feedback, the multiple one exhibits resonances between feedback and nonlinear harmonics, leading to a resonantly strong or weak oscillation coherence even for a small anharmonicity. Analytical results are confirmed numerically for van der Pol and van der Pol-Duffing oscillators.

Abstract: In the absence of fractures, methane bubbles in deep-water sediments are immovably trapped within a porous matrix by surface tension. The dominant mechanism of transfer of gas mass therefore becomes the diffusion of gas molecules through porewater. The accurate description of this process requires non-Fickian diffusion to be accounted for, including both thermodiffusion and gravitational action. We evaluate the diffusive flux of aqueous methane considering non-Fickian diffusion and predict the existence of bubble mass accumulation zones within deep-water sediments. The occurrence of these may be highly significant in the assessment of methane hydrate reservoirs or inventories as they could occur independently of the hydrate stability zone, yet may mimic the bottom-simulating-reflector which is commonly used to identify the base of the zone.

Abstract: Porous sediments in geological systems are exposed to stress by the above-lying mass and consequent compaction, which may be significantly nonuniform across the massif. We derive scaling laws for the compaction of sediments of similar geological origin. With these laws, we evaluate the dependence of the transport properties of a fluid-saturated porous medium (permeability, effective molecular diffusivity, hydrodynamic dispersion, electrical and thermal conductivities) on its porosity. In particular, we demonstrate that the assumption of a uniform geothermal gradient is not adequate for systems with nonuniform compaction and show the importance of the derived scaling laws for mathematical modelling of methane hydrate deposits; these deposits are believed to have potential for impact on global climate change and glacial-interglacial cycles.

Abstract: We consider a liquid bearing gas bubbles in a porous medium. When gas bubbles are immovably trapped in a porous matrix by surface tension forces, the dominant mechanism of transfer of gas mass becomes the diffusion of gas molecules through the liquid. Essentially, the gas solution is in local thermodynamic equilibrium with vapor phase all over the system, i.e., the solute concentration equals the solubility. When temperature and/or pressure gradients are applied, diffusion fluxes appear and these fluxes are faithfully determined by the temperature and pressure fields, not by the local solute concentration, which is enslaved by the former. We derive the equations governing such systems, accounting for thermodiffusion and gravitational segregation effects which are shown not to be neglected for geological systems-marine sediments, terrestrial aquifers, etc. The results are applied for the treatment of non-high pressure systems and real geological systems bearing methane or carbon dioxide, where we find a potential possibility of the formation of gaseous horizons deep below a porous medium surface. The reported effects are of particular importance for natural methane hydrate deposits and the problem of burial of industrial production of carbon dioxide in deep aquifers.

Abstract: An effective white-noise Langevin equation is derived that describes long-time phase dynamics of a limit-cycle oscillator driven by weak stationary colored noise. Effective drift and diffusion coefficients are given in terms of the phase sensitivity of the oscillator and the correlation function of the noise, and are explicitly calculated for oscillators with sinusoidal phase sensitivity functions driven by two typical colored Gaussian processes. The results are verified by numerical simulations using several types of stochastic or chaotic noise. The drift and diffusion coefficients of oscillators driven by chaotic noise exhibit anomalous dependence on the oscillator frequency, reflecting the peculiar power spectrum of the chaotic noise.

Abstract: A recent paper claims that mean characteristics of chaotic orbits differ from the corresponding values averaged over the set of unstable periodic orbits, embedded in the chaotic attractor. We demonstrate that the alleged discrepancy is an artifact of the improper averaging. Since the natural measure is nonuniformly distributed over the attractor, different periodic orbits make different contributions into the time averages. As soon as the corresponding weights are accounted for, the discrepancy disappears.

Abstract: The phase description is a powerful tool for analyzing noisy limit-cycle oscillators. The method, however, has found only limited applications so far, because the present theory is applicable only to Gaussian noise while noise in the real world often has non-Gaussian statistics. Here, we provide the phase reduction method for limit-cycle oscillators subject to general, colored and non-Gaussian, noise including a heavy-tailed one. We derive quantifiers like mean frequency, diffusion constant, and the Lyapunov exponent to confirm consistency of the results. Applying our results, we additionally study a resonance between the phase and noise.

Abstract: Frozen parametric disorder can lead to the appearance of sets of localized convective currents in an otherwise stable (quiescent) fluid layer heated from below. These currents significantly influence the transport of an admixture (or any other passive scalar) along the layer. When the molecular diffusivity of the admixture is small in comparison to the thermal one, which is quite typical in nature, disorder can enhance the effective (eddy) diffusivity by several orders of magnitude in comparison to the molecular diffusivity. In this paper, we study the effect of an imposed longitudinal advection on the delocalization of convective currents, both numerically and analytically, and report a subsequent drastic boost of the effective diffusivity for weak advection.

Abstract: We study transport of a weakly diffusive pollutant (a passive scalar) through thermoconvective flow in a fluid-saturated horizontal porous layer heated from below under frozen parametric disorder. In the presence of disorder (random frozen inhomogeneities of the heating or of macroscopic properties of the porous matrix), spatially localized flow patterns appear below the convective instability threshold of the system without disorder. Thermoconvective flows crucially affect the transport of a pollutant along the layer, especially when its molecular diffusion is weak. The effective (or eddy) diffusivity also allows us to observe the transition from a set of localized currents to an almost everywhere intense �global� flow. We present results of numerical calculation of the effective diffusivity and discuss them in the context of localization of fluid currents and the transition to a �global� flow. Our numerical findings are in good agreement with the analytical theory that we develop for the limit of a small molecular diffusivity and sparse domains of localized currents. Though the results are obtained for a specific physical system, they are relevant for a broad variety of fluid dynamical systems.

Abstract: We discuss the problem of proteasomal degradation of proteins. Though proteasomes are important for all aspects of cellular metabolism, some details of the physical mechanism of the process remain unknown. We introduce a stochastic model of the proteasomal degradation of proteins, which accounts for the protein translocation and the topology of the positioning of cleavage centers of a proteasome from first principles. For this model we develop a mathematical description based on a master equation and techniques for reconstruction of the cleavage specificity inherent to proteins and the proteasomal translocation rates, which are a property of the proteasome species, from mass spectroscopy data on digestion patterns. With these properties determined, one can quantitatively predict digestion patterns for new experimental set-ups. Additionally we design an experimental set-up for a synthetic polypeptide with a periodic sequence of amino acids, which enables especially reliable determination of translocation rates.

Abstract: In a range of physical systems, the first instability in Rayleigh-Bérnard convection between nearly thermally insulating horizontal plates is large scale. This holds for thermal convection of fluids saturating porous media. Large-scale thermal convection in a horizontal layer is governed by remarkably similar equations both in the presence of a porous matrix and without it, with only one additional term for the latter case, which, however, vanishes under certain conditions (e.g., two-dimensional flows or infinite Prandtl number). We provide a rigorous derivation of long-wavelength equations for a porous layer with inhomogeneous heating and possible pumping.

Abstract: For noisy self-sustained oscillators, both reliability, the stability of a response to a noisy driving, and coherence, understood in the sense of constancy of oscillation frequency, are important characteristics. Although both characteristics and techniques for controlling them have received great attention from researchers, owing to their importance for neurons, lasers, clocks, electric generators, etc., these characteristics were previously considered separately. In this paper, a strong quantitative relation between coherence and reliability is revealed for a limit cycle oscillator subject to a weak noisy driving and a linear delayed feedback, a convection control tool. The analytical findings are verified and enriched with a numerical simulation for the Van der Pol-Duffing oscillator.

Abstract: Soret-driven convection of a binary mixture in a shallow porous layer is analyzed. The analysis focuses on the behavior of the system in the presence of a concentration or heat source. In the long-wavelength limit, regimes are found in which the flow regions near the source and at the periphery are separated by narrow annular transition regions. It is also shown that the outward concentration flux from the source is dominated by convection, whereas heat can be transferred from the source both by convection and by diffusion. Multistability between these two regimes is possible.

Abstract: We develop a weakly nonlinear theory of the Kuramoto transition in an ensemble of globally coupled oscillators in presence of additional time-delayed coupling terms. We show that a linear delayed feedback not only controls the transition point, but effectively changes the nonlinear terms near the transition. A purely nonlinear delayed coupling does not effect the transition point, but can reduce or enhance the amplitude of collective oscillations.

Abstract: We demonstrate, within the framework of the FitzHugh-Nagumo model, that a firing neuron can respond to a noisy driving in a nonreliable manner: the same Gaussian white noise acting on identical neurons evokes different patterns of spikes. The effect is characterized via calculations of the Lyapunov exponent and the event synchronization correlations. We construct a theory that explains the antireliability as a combined effect of a high sensitivity to noise of some stages of the dynamics and nonisochronicity of oscillations. Geometrically, the antireliability is described by a random noninvertible one-dimensional map.

Abstract: We study the stability of self-sustained oscillations under the influence of external noise. For small-noise amplitude a phase approximation for the Langevin dynamics is valid. A stationary distribution of the phase is used for an analytic calculation of the maximal Lyapunov exponent. We demonstrate that for small noise the exponent is negative, which corresponds to synchronization of oscillators.

Abstract: We consider the effect of external noise on the dynamics of limit cycle oscillators. The Lyapunov exponent becomes negative under influence of small white noise, what means synchronization of two or more identical systems subject to common noise. We analytically study the effect of small non-identities in the oscillators and in the noise, and derive statistical characteristics of deviations from the perfect synchrony. Large white noise can lead to desynchronization of oscillators, provided they are non-isochronous. This is demonstrated for the Van der Pol-Duffing system.

Abstract: A theory of effect of delayed feedback on the coherence of the noisy self-sustained oscillations is developed. In the Gaussian approximation a closed system of equations is derived for the phase diffusion constant and the mean frequency. A comparison with numerics shows that the theory works well for weak feedback and strong noise.

Abstract: We demonstrate that the coherence of a noisy or chaotic self-sustained oscillator can be efficiently controlled by the delayed feedback. We develop a theory of this effect, considering noisy systems in the Gaussian approximation. We obtain a closed equation system for the phase diffusion constant and the mean frequency of oscillation. For weak feedback and strong noise, the theory is in good agreement with the numerics. We discuss possible applications of the effect for the synchronization control.

Abstract: We consider a population of parametrically excited globally coupled oscillators in a weakly nonlinear state. The instabilities of collective modes lead to a traveling-wave regime, where intensities of oscillations of each oscillator vary periodically in time. For large excitation amplitudes a frozen state with nearly uniform oscillation intensities is observed.

Abstract:

Abstract: The dissertation contains 4 original chapters: <br> 1) Parametric excitation of Soret-driven convection of binary mixture in a horizontal porous layer. <br> 2) Soret-driven convection in a horizontal porous layer from a heat or concentration source. <br> 3) Localization of convective flows under randomly inhomogeneous heating. <br> 4) Synchrony of nonlinear systems driven by common noise.

Abstract: In the present dissertation paper we study problems related to synchronization phenomena in the presence of noise which unavoidably appears in real systems. One part of the work is aimed at investigation of utilizing delayed feedback to control properties of diverse chaotic dynamic and stochastic systems, with emphasis on the ones determining predisposition to synchronization. Other part deals with a constructive role of noise, i.e. its ability to synchronize identical self-sustained oscillators. <br><br> First, we demonstrate that the coherence of a noisy or chaotic self-sustained oscillator can be efficiently controlled by the delayed feedback. We develop the analytical theory of this effect, considering noisy systems in the Gaussian approximation. Possible applications of the effect for the synchronization control are also discussed. <br><br> Second, we consider synchrony of limit cycle systems (in other words, self-sustained oscillators) driven by identical noise. For weak noise and smooth systems we proof the purely synchronizing effect of noise. For slightly different oscillators and/or slightly nonidentical driving, synchrony becomes imperfect, and this subject is also studied. Then, with numerics we show moderate noise to be able to lead to desynchronization of some systems under certain circumstances. For neurons the last effect means �antireliability� (the �reliability� property of neurons is treated to be important from the viewpoint of information transmission functions), and we extend our investigation to neural oscillators which are not always limit cycle ones. <br><br> Third, we develop a weakly nonlinear theory of the Kuramoto transition (a transition to collective synchrony) in an ensemble of globally coupled oscillators in presence of additional time-delayed coupling terms. We show that a linear delayed feedback not only controls the transition point, but effectively changes the nonlinear terms near the transition. A purely nonlinear delayed coupling does not affect the transition point, but can reduce or enhance the amplitude of collective oscillations.