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We consider a nonlinear 6-dimensional dynamic system which is a model of functioning of one simple molecular repressilator and find sufficient conditions of existence of a cycle C in the phase portrait of this system. An invariant neighborhood of C which retracts to C has been constructed.

This article analyzes the current status of professional education of future teachers-artists in the context of changing educational paradigm. The authors consider the specifics, the current state of training of future teachers of fine arts. The authors highlight the problems and outline the trends of the system of professional development of undergraduate students. Basic organizational and pedagogical condition is determined as the integration of scientific knowledge, experience of creative activities and special training in the field of pedagogy and psychology. In general, the process is seen as based on science-based innovative pedagogical technologies.

The article deals with the specifics of training the masters to crafts in the first third of the twentieth century, which were linked to a new State policy. We describe its economic premise, which was that traditionally in the middle zone of Russian State handicrafts and their fabrication have made substantial contributions to the income of a peasant. The authors draw attention to the fact that industrial approach, with its samples, bulk copy system has also a negative influence on the artistic side of the manufactured products. In the work it is claimed that the greatest harm that crafts and decorative-applied arts development suffered was from industrial attitudes towards the manufactured products that ultimately led to the closure of many original and interesting handicraft workshops in the 1930-ies. Based on historical analysis, the important role of folk art in the organization of cottage graft industry is emphasized. The article reveals the positive and negative sides of artisanal workshops, their rebirth in folk art crafts enterprises. We describe three levels of the system of training artists in the form of: training and demonstration workshops, training schools, technical schools and museums of cottage craft industry, the concept of organization of instructor schools and their obligations. The authors conclude that the instructor of the school, and the entire system of training for crafts was a prototype artist-teacher training system for schools and after-school educational institutions. Instructor schools comply with today's feature art colleges, art schools and clubs and electives in applied art. The fact is underlined that some negative points that emerged in the work of training schools, the attitude toward them, moved into the modern system of additional education and conducting classes for the applied arts. At the same time, a lot of positive experience proved to be forgotten and can be successfully applied in modern training of professionals in the field of arts and crafts.

The article is devoted to revealing the potential of the basic disciplines in the development of professional competence of future teachers of the fine arts on the example of academic drawing, painting and academic composition. The authors believe that the study of these basic disciplines provides, first of all, provides fundamental training in the development of professional skills of the future teacher of the fine arts. Although the appropriate professional level is formed throughout the entire period of professional activity, already during the student years the foundations are laid of what is called «learning to do», i.e. certain level of professional artistry. This is, in fact, one of the aspects of the professional competence of the future teacher of the fine arts.

This article is devoted to the analysis and study of modern problems of additional mathematical education, and also to the research of further ways of developing of this field of education. The authors consider the already existing system of mathematical education and trace influence of the traditions of the Soviet era on the current picture of this kind of education. It is noteworthy that in the article mathematics is regarded not only as a discipline and subject of teaching, but also as an educational practice that can be represented in different forms. The existing forms of additional mathematical education are exposed to the critical analysis, their effective components and deficits are distinguished. Such forms as clubs for mathematics lovers, the subject-specific mathematical olympiads (competitions), specialized physical and mathematical schools, etc. are considered. Special attention is paid to summer schools. The focus is on the practice of selection and support of mathematically gifted students as having a key influence on the construction of mathematical education in the XX century as well as now. Besides the analysis of the bases of the existing approaches and purposes of mathematical education, modern contexts and priorities of mathematical education following from them are studied. Prospects for mathematical education of such formats as games and simulators (including computer games), TV series (as a form of promoting some specific knowledge), etc. are analyzed. The authors set forth a hypothesis concerning the vectors within which an expansion of mathematical additional education should occur. The paper attempts to answer the following question: what kinds of new formats should appear in the field of mathematics education; with what popular media trends this sphere could be synthesized; in what modern public practical circumstances the specificity of mathematical knowledge can be applied. In particular, the formats of intensive schools, elective courses, clubs and the variety of their types are considered as perspective and promising.

The purpose of this article is to analyze the problems of formation of professional orientation of the education of students on the basis of studying the graphic disciplines. Graphical presentation of information facilitates visualization and modeling of various processes in the study of theoretical and practical material in the process of studying technical subjects. A special role is played by the language of graphics in creating the information and communication environment in engineering education. Graphic training as a component of engineering education should perform the following interrelated functions: the educational, technical, special, and general cultural ones.

This article focuses upon special aspects of teaching students of the Department of Foreign Languages to construct teacher-made tests for effective application at school. Teacher-made tests estimate the pupil achievements. Their objectivity improves the learners’ development. The mathematical-statistical measurement of test results, including reliability and validity, is used to the foundation of objectivity of the procedure. The article emphasizes the professional orientation of learning process, a new vision of assessment materials according to modern trends of pedagogy. We offer a curriculum for teaching the test theory, peculiar aspects of language tests, as well as their role in forming the communicative competence. It is important to understand what can be tested, what different tests measure, including cloze-tests. Practical classes are held to form students’ skills to construct tests and estimate the items statistical characteristics using computer program.

The article focuses on the role of the Museum space in the preservation of cultural and historical heritage of humanity. Museum education is a new field of educational activity and can be considered as an innovative pedagogical technology. Modern innovative technologies improve the process of perception of Museum information through the creation of the effect of belonging. The development of interactive technology promotes a more intimate communication of the Museum visitor with the exposition. In recent years there has been quite a different understanding of the relationship between Museum and visitor, established the new term of «cultural-educational activity», which involves education in the cultural space. Integrated use of socio-cultural opportunities cultural and educational organization provides an effective interaction of traditional and innovative methods of education - the revival of the historical continuity of culture based on national traditions, the reconstruction of the largely lost system of holidays, rituals, traditional forms of entertainment, the renewed interest in folk art, crafts, folk games, development of creative potential of personality, patriotism and citizenship.

The article is devoted to socio-pedagogical problems of overcoming of deviant behavior of children and adolescents by means of social tourism in the Orenburg region. The category of consumers of tourist services is children and adolescents who are in difficult life situation characterized by deviant behavior. The paper describes the importance and the need to achieve educational and social goals - the formation of the spiritual foundations of children and adolescents in the traditions of the Orenburg Cossacks by means of social tourism. The author of the article revealed the specific features of spiritual and material culture of the Orenburg Cossacks, which is theoretical-methodological basis of technology of upbringing and education of children and adolescents, characterized by deviant behavior; the author proved that social tourism is a powerful tool and an effective pedagogical form for the formation of positive models of behavior of children and adolescents.