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## Optimal control in the New Keynesian model with monetary and fiscal policy interactions

Dynamics of the New Keynesian model in continuous time with the Rotemberg pricing mechanism is considered within a framework of an optimal control problem. Various regimes of monetary and fiscal policy ('active' and 'passive') can lead to unstable dynamics in the economy. Parameters of the Taylor rules for both monetary and fiscal policies determine conditions for local equilibrium determinacy. Mapping out the ranges of the Taylor coefficient values where local determinacy cannot be obtained allows to control the economic system by controlling these parameters

The approaches to the modeling of innovative development of economic systems based on the methodology of nonlinear dynamics are proposed. The generalized nonlinear dynamic models for the analysis of economic development is discussed. The loss of stability of the dynamic mode and the area of deterministic chaos are considered in terms of risk analysis

Main concepts and models of the modern theory of self-organization of complex systems, called also synergetics, are generalized and formulated in the book as principles of a synergetic world view. They are under discussion in the context of philosophical studies of holism, teleology, evolutionism as well as of gestalt-psychology; they are compared with some images from the history of human culture. The original and unfamiliar (to the Western readers) research results of the Moscow synergetic school which has its center at the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences are expounded in the book. Complicated and paradoxical concepts of synergetics (structure-attractors, bifurcations, blow-up regimes, non-stationary dissipative structures of self-organization, fractals, non-linearity) are translated into an intelligible language and vividly illustrated by materials and examples from various fields of knowledge, starting with the laser thermonuclear fusion and concluding with mysterious phenomena of human psychology and creativity. The style of writing is close to that of popular-science literature. That's why the book might be of interest and is quite comprehensible for students and specialists in the humanities. It is shown that the development of synergetics entails deep changes in the conceptual net through which we comprehend the world. It means a radical shift of paradigm, a conceptual transition from being to becoming, from stability to sustainability, from images of order to chaos generating new ordered evolving structures, from self-maintaining systems to fast evolution through a nonlinear positive feedback, from evolution to co-evolution, reciprocal evolution of different complex systems. The new synergetic way of thinking is evolutionary, nonlinear and holistic. This is a modern stage of development within the traditions of cybernetics and system-structural analysis. However, many elements of the latter have undergone important changes since their appearance.

Some texts written by me together with corresponding member of the Russian Academy of science Sergei P. Kurdyumov (1928-2004) and under his direct ideological influence are collected in the book. These texts are elaborated, systematized, brought together in the book and supplemented with new materials. Sergei P. Kurdyumov were possessed of a deep metaphysical flair and put forward ideas, the matter of which are not fully clear up to now. These are, first of all, the idea of co-evolution and the notion of complex structures developing at different tempos as co-existing tempo-worlds. Owing to developments in the field of nonlinear dynamics and of synergetics, the classical problem of time and the problems of evolutionary holism disclose some new and non-traditional aspects. The matter of new notions of nonlinearity of the course of time in the processes of evolution and coevolution and of nonlinear links between different modi of time – between the past, the present and the future - come to the light in the book. Analyses of four interconnected aspects of the course of processes in open and nonlinear dissipative systems – of evolutionarity, temporality, emergent nature and holism – are carried out. A whole series of paradoxical notions, such as “the influence of the future upon the present”, “the possibility of touch of a remote future in praesenti”, irreversibility and elements of reversibility of the course of time appear in synergetics, non-traditional and nonlinear notion of time being at the heart of all of them. It is shown that the best pictorial view of the nonlinear time is apparently the tree of evolution or the tree of time that represent one of archetypes in the human psyche. This image is widely used in myths and religious doctrines of the world nations (the tree of evolution of languages from some united parent language or the tree of evolution of biological species), the image is often drawn by children, appears in consciousness of a man in his sleep, etc. The synergetics methodology under development is applied to study of cognitive systems. The emergent structures of evolution and of self-organization of the individual consciousness, their spatiotemporal peculiarities, and the complexity of the human Self are considered in detail. The radical changes in the understanding of the problems of time that occur due to synergetics are compared with images of time and with the notions of connection between the past, the present and the future in the history of philosophy and of culture. The obtained methodological inferences are of great importance for a reform of systems of education, for forecasting (for building of scenarios of future development), for effective management activity in the modern globalizing world, for elaboration of methods of stimulation of the creative thinking, for the growth of personality and its adequate building into the social media.

The dynamic approach to understanding of the human consciousness, its cognitive activities and cognitive architecture is one of the most promising approaches in the modern epistemology and cognitive science. The conception of embodied mind is under discussion in the light of nonlinear dynamics and of the idea co-evolution of complex systems developed by the Moscow scientific school. The cognitive architecture of the embodied mind is rather complex: data from senses and products of rational thinking, the verbal and the pictorial, logic and intuition, the analytical and synthetic abilities of perception and of thinking, the local and the global, the analogue and the digital, the archaic and the post-modern are intertwined in it. In the process of cognition, co-evolution of embodied mind as an autopoietic system and its surroundings takes place. The perceptual and mental processes are bound up with the structure of human body. Nonlinear and circular connecting links between the subject of cognition and the world constructed by him can be metaphorically called a nonlinear cobweb of cognition. Cognition is an autopoietic activity because it is directed to the search of elements that are missed; it serves to completing integral structures. According to the theory of blow-up regimes in complex systems elaborated by Sergey P.Kudyumov and his followers, the idea of co-evolution is connected with the concept of tempoworlds. To co-evolve means to start to develop in one and the same tempoworld and to use the possibility – in case of a proper intergation into a whole structure – to accelerate the tempo of evolution. The cognitive activities of the human being can be considered as a movement (active walk) in landscapes of co-evolution when he cognizes and changes environment and is changed himself by the very activities. The similar conclusion can be drawn from Francisco Varela’s conception of enactive cognition.

This article discusses examples of nonlinear models of economic dynamics and possibilities of their research by numerical procedures in MATLAB. Demonstrated specific effects of these models, in particular, the possibility of forming a chaotic behavior

We propose a modification of the Crow-Kimura and Eigen models of biological molecular evolution to include a mutator gene that causes both an increase in the mutation rate and a change in the fitness landscape. This mutator effect relates to a wide range of biomedical problems. There are three possible phases: mutator phase, mixed phase and non-selective phase. We calculate the phase structure, the mean fitness and the fraction of the mutator allele in the population, which can be applied to describe cancer development and RNA viruses. We find that depending on the genome length, either the normal or the mutator allele dominates in the mixed phase. We analytically solve the model for a general fitness function. We conclude that the random fitness landscape is an appropriate choice for describing the observed mutator phenomenon in the case of a small fraction of mutators. It is shown that the increase in the mutation rates in the regular and the mutator parts of the genome should be set independently; only some combinations of these increases can push the complex biomedical system to the non-selective phase, potentially related to the eradication of tumors.

The classical cybernetics in the Norbert Wiener’s tradition is nowadays a part of the mathematical theory of complex systems and nonlinear dynamics. Only in these frameworks, building of structures and patterns in nature and technics can be explained and in computer models simulated. Self-organization and emergence became welldefined concepts and can be transferred to technical systems. In the first part of the article, the foundations of complex systems and of nonlinear dynamics are under review. As an application, the building of structures and patterns in complex cell systems, which are subject of system biology, is considered. In the second part, the application of complex system dynamics to evolution of brain and cognition is explored. The research gives us a prerequisite for development of cognitive and social robots, what the topic of the third part is. Neural network structures are not at all limited to individual organisms and robots. In the fourth part, the cyberphysical systems, by means of which complex self-controlling sociotechnical systems are modeled, are studied. The mathematical theory of complex systems and nonlinear dynamics provides us with foundation for understanding of self-organization and emergence in this field. Finally, the question of ethical and social general conditions for technical constructing of complex self-organizing systems are stated and discussed.

Main concepts and models of the modern theory of self-organization of complex systems, called also synergetics, are generalized and formulated in the book as principles of a synergetic world view. They are under discussion in the context of philosophical studies of holism, teleology, evolutionism as well as of gestalt-psychology; they are compared with some images from the history of human culture. The original and unfamiliar (to the Western readers) research results of the Moscow synergetic school which has its center at the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences are expounded in the book. The heuristic value of the synergetic models of evolution and self-organization of complex systems in epistemology and cognitive psychology, education and teaching, futures studies, social management activities and systems of global security is shown in the book. The book is addressed to a wide circle of readers: students, teachers, scientists who are specialized in different fields of natural sciences and the humanities as well as to all readers who strive for using recent results of science for reflections and achieving success in their own life.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.