Typicalness of chaotic fractal behaviour of integral vortexes in Hamiltonian systems with discontinuous right hand side

We consider a linear-quadratic deterministic optimal control problem where the control takes values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely the chaotic behaviour of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighbourhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us the possibility to calculate the entropy and the Hausdorff dimension of the non-wandering set which appears to have a Cantor-like structure as in Smale's Horseshoe. The dynamics of the system is described by a topological Markov chain. In the second part it is shown that this behaviour is generic for piece-wise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.Comment: 113 pages, 22 figure

Nonlinear Stochastic Models for Water Level Dynamics in Closed Lakes

This paper presents the results of investigation of nonlinear mathematical models of the behavior of closed lakes using the example of the Caspian Sea. Forecasting the level of the Caspian Sea is crucial both for the economy of the region and for the region's environment. The Caspian Sea is a closed reservoir; it is well known that its level changes considerably due to a variety of factors including global climate change. A series of forecasts exists based on different methods and taking into account some of the following factors: the influence of the sun's activity; the atmospheric circulation; the changing shape of the world's ocean; geological phenomena; the river inflow; and the velocity of evaporation. All of these models were calculated based on the linearization of the relations considered. For the last two decades, the most popular model has been the linear stochastic equation of water balance. This model was used as the base of the well known project of reversing the flow of the northward-flowing rivers. But the real behavior of the Caspian Sea contradicted the forecasting done using this model. One of the reasons of the failure was ignorance of the relations mentioned above. We are inclined to think however that the main reason for failure was that the forecast used a linear equation. The goal of the present paper is to analyze and generalize, from the modern mathematical point of view, the forecasting methodology for the level of the Caspian Sea, including the nonlinear effects crucial influence on the dynamics of sea level. In particular, the mathematical problems concerning the nonlinear stochastic equations are considered

Localized and controlled delivery of nitric oxide to the conventional outflow pathway via enzyme biocatalysis: towards therapy for Glaucoma

Nitric oxide (NO) has been shown to lower intraocular pressure (IOP), however its therapeutic effects on outflow physiology are location- and dose-dependent. Here, a NO delivery platform that directly targets the resistance-generating region of the conventional outflow pathway and locally liberates a controlled dose of NO is reported. An increase in outflow facility (decrease in IOP) is demonstrated in mouse model

Enzyme prodrug therapy achieves site-specific, personalized physiological responses to the locally produced nitric oxide

Nitric oxide (NO) is a highly potent but short-lived endogenous radical with a wide spectrum of physiological activities. In this work, we developed an enzymatic approach to the site-specific synthesis of NO mediated by biocatalytic surface coatings. Multilayered polyelectrolyte films were optimized as host compartments for the immobilized β-galactosidase (β-Gal) enzyme through a screen of eight polycations and eight polyanions. The lead composition was used to achieve localized production of NO through the addition of β-Gal–NONOate, a prodrug that releases NO following enzymatic bioconversion. The resulting coatings afforded physiologically relevant flux of NO matching that of the healthy human endothelium. The antiproliferative effect due to the synthesized NO in cell culture was site-specific: within a multiwell dish with freely shared media and nutrients, a 10-fold inhibition of cell growth was achieved on top of the biocatalytic coatings compared to the immediately adjacent enzyme-free microwells. The physiological effect of NO produced via the enzyme prodrug therapy was validated ex vivo in isolated arteries through the measurement of vasodilation. Biocatalytic coatings were deposited on wires produced using alloys used in clinical practice and successfully mediated a NONOate concentration-dependent vasodilation in the small arteries of rats. The results of this study present an exciting opportunity to manufacture implantable biomaterials with physiological responses controlled to the desired level for personalized treatment