On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].Comment: 35 pages, 4 figures, PDFLaTe

Stability of Nonlinear Normal Modes in the FPU-β\beta Chain in the Thermodynamic Limit

All possible symmetry-determined nonlinear normal modes (also called by simple periodic orbits, one-mode solutions etc.) in both hard and soft Fermi-Pasta-Ulam-$\beta$ chains are discussed. A general method for studying their stability in the thermodynamic limit, as well as its application for each of the above nonlinear normal modes are presented

q-Breathers and thermalization in acoustic chains with arbitrary nonlinearity index

Nonlinearity shapes lattice dynamics affecting vibrational spectrum, transport and thermalization phenomena. Beside breathers and solitons one finds the third fundamental class of nonlinear modes -- $q$-breathers -- periodic orbits in nonlinear lattices, exponentially localized in the reciprocal mode space. To date, the studies of $q$-breathers have been confined to the cubic and quartic nonlinearity in the interaction potential. In this paper we study the case of arbitrary nonlinearity index $\gamma$ in an acoustic chain. We uncover qualitative difference in the scaling of delocalization and stability thresholds of $q$-breathers with the system size: there exists a critical index $\gamma^*=6$, below which both thresholds (in nonlinearity strength) tend to zero, and diverge when above. We also demonstrate that this critical index value is decisive for the presence or absense of thermalization. For a generic interaction potential the mode space localized dynamics is determined only by the three lowest order nonlinear terms in the power series expansion.Comment: 5 pages, 4 figure

Дослідження гострої токсичності топічної комбінації з глюкозаміном і кетопрофеном у формі крем-гелю

Safety of new drugs is the most important criteria for the study, and it is determined by the analysis of acute toxicity. When determining acute toxicity the adverse effects of drugs are described in their single use or multiple administrations in a short period of time.Aim. To determine the toxicity class of a new topical combination containing 5.0 % glucosamine hydrochloride and 2.0 % in the form of a cream-gel or cutaneous application and intragastric introduction.Materials and methods. The studies were carried out in accordance with EC Directive 86/609 EEC. The cutaneous application of the combination was in the range of doses from 43 to 22600 mg/kg. The object studied was used in the dose range from 500 to 5000 mg/kg in intragastric introduction. The behavior of animals and their survival were observed within 14 days.Results. During the observation after cutaneous application there was no mortality, changes in the common life cycle of rats and the skin irritation. In intragastric introduction of the combination its LD50 was higher than 5000 mg/kg.Conclusions. Therefore, the study of acute toxicity of the combination has shown the absence of mortality in rats at the maximum allowable doses for cutaneous application and intragastric introduction. This fact indicates a low toxicity of the G/K cream-gel, and it gives the possibility to refer the combination studied to the category of relatively harmless substances.Введение. Безопасность новых лекарственных средств является наиболее важным критерием для исследования и определяется с помощью анализа острой токсичности. При постановке острой токсичности описываются побочные эффекты лекарственных средств при их однократном воздействии или множественных введениях в течение короткого периода времени.Цель исследования. Целью данного исследования стало определение класса токсичности новой топической комбинации, содержащей 5,0 % глюкозамина гидрохлорида и 2,0 % кетопрофена в форме крем-геля при накожной аппликации и внутрижелудочном введении.Материалы и методы. Все исследования были проведены в соответствии с директивой ЕС 86/609 ЕЕС. Накожная аппликация комбинации соответствовала диапазону доз от 43 до 22600 мг/кг. При внутрижелудочном введении исследуемый объект применялся в диапазоне доз от 500 до 5000 мг/кг. Выживание животных и их поведение фиксировали на протяжении 14 дней.Результаты. При наблюдении после накожного применения комбинации не было зафиксировано летальности среди животных, изменений в общем жизненном цикле крыс и накожных высыпаний. При внутрижелудочном введении комбинации ее LD50 была больше чем 5000 мг/кг.Выводы. Следовательно, исследование острой токсичности комбинации показало отсутствие летальности животных при максимально допустимых дозах для ее накожной аппликации и внутрижелудочного введения. Этот факт свидетельствует о низкой токсичности исследуемой комбинации и дает возможность отнести ее к категории относительно безопасных веществ.Безпека нових лікарських засобів є найбільш важливим критерієм для дослідження і визначається за допомогою аналізу гострої токсичності. При постановці гострої токсичності описуються побічні ефекти лікарських засобів при їх одноразовому або множинних введеннях впродовж короткого періоду часу.Мета дослідження. Метою даного дослідження стало визначення класу токсичності нової топічної комбінації, що містить 5,0 % глюкозаміну гідрохлориду та 2,0 % кетопрофену у формі крем-гелю при нашкірній аплікації і внутрішньошлунковому введенні.Матеріали та методи. Всі дослідження були проведені відповідно до директиви ЄС 86/609 ЕЕС. Нашкірна аплікація комбінації відповідала діапазону доз від 43 до 22600 мг/кг. При внутрішньошлунковому введенні досліджуваний об’єкт застосовувався в діапазоні доз від 500 до 5000 мг/кг. Виживання тварин і їх поведінка були зафіксовані впродовж 14 діб.Результати. При спостереженні після зовнішнього застосування комбінації не було відмічено летальності серед тварин, змін у загальному життєвому циклі щурів і нашкірних висипань. При внутрішньошлунковому введенні комбінації її LD50 було більше за 5000 мг/кг.Висновки. Отже, дослідження гострої токсичності комбінації показало відсутність летальності тварин при максимально допустимих дозах для її нашкірної аплікації і внутрішньошлункового введення. Цей факт свідчить про низьку токсичність досліджуваної комбінації і дає можливість віднести її до категорії відносно нешкідливих речовин

Stable Exact Solutions in Cosmological Models with Two Scalar Fields

The stability of isotropic cosmological solutions for two-field models in the Bianchi I metric is considered. We prove that the sufficient conditions for the Lyapunov stability in the Friedmann-Robertson-Walker metric provide the stability with respect to anisotropic perturbations in the Bianchi I metric and with respect to the cold dark matter energy density fluctuations. Sufficient conditions for the Lyapunov stability of the isotropic fixed points of the system of the Einstein equations have been found. We use the superpotential method to construct stable kink-type solutions and obtain sufficient conditions on the superpotential for the Lyapunov stability of the corresponding exact solutions. We analyze the stability of isotropic kink-type solutions for string field theory inspired cosmological models.Comment: 23 pages, v3:typos corrected, references adde

Influence of a high-frequency pulsed nanosecond diffusion discharge in the nitrogen atmosphere on the electrical characteristics of a CdHgTe epitaxial films

The effect of a high-frequency nanosecond volume discharge forming in an inhomogeneous electrical field at atmospheric pressure on the CdHgTe (CMT) epitaxial films is studied. The measurement of the electrophysical parameters of the CMT specimens upon irradiation shows that that the action of pulses of nanosecond volume discharge leads to changes in the electrophysical properties of CMT epitaxial films due to formation of a near-surface high-conductivity layer of the n-type conduction. The preliminary results show that it is possible to use such actions in the development of technologies for the controlled change of the properties of CMT narrow-band solid solutions and production of structures heterogeneous with respect to conduction

BRIEF CHARACTERISTIC OF EUROPEAN GENOTYPE TICK-BORNE ENCEPHALITIS VIRUS STRAINS IDENTIFIED IN SIBERIAN REGION

The molecular-genetic analysis of 13 strains of Western genotype TBEV isolated in Western and Eastern Siberia demonstrated two groups of strains differed geneticallyfrom each other and had a high level of E gene sequences homology within each group. Nevertheless, the heterogeneity of biological propertiesfor some strains within a group was observed

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function