Cybernetics and Systems Analysis, Vol. 52, No . 3, May, 2016 SYSTEMS ANALYSIS DIFFERENTIAL–ALGEBRAIC EQUATIONS AND DYNAMIC SYSTEMS ON MANIFOLDS To the cherished memory of Academician A. I. Kukhtenko Iu. G. Kryvonos, 1 V. P. Kharchenko, 2† and N. M. Glazunov 2‡ UDC 681.5+513.6+517.9 Abstract. The authors consider current problems of the modern theory of dynamic systems on manifolds, which are actively developing. A brief review of such trends in the theory of dynamic systems is given. The results of the algebra of dual numbers, quaternionic algebras, biquaternions (dual quaternions), and their application to the analysis of infinitesimal neighborhoods and infinitesimal deformations of manifolds (schemes) are presented. The theory of differential–algebraic equations over the field of real numbers and their dynamics, as well as elements of trajectory optimization of respective dynamic systems, are outlined. On the basis of connection in bundles, the theory of differential–algebraic equations is extended to algebraic manifolds and schemes over arbitrary fields and schemes, respectively. Keywords: dual number, dual quaternion, quaternion algebra, algebraic manifold, scheme, deformation, differential–algebraic equation, mathematical model, dynamic system, differential equation on algebraic manifold. INTRODUCTION In his studies in mechanics, control theory, and systems theory, Academician A. I. Kuhtenko analyzed, applied, and popularized dual numbers, biquaternions, differential–algebraic equations, dynamic systems on manifolds, toposes, and theory of categories and functors [1–4]. In reports on seminars and comments therein, he mentioned that along with systemic studies, it is necessary to study and use new mathematical structures and corresponding methods. Kukhtenko also initiated and supervised other fundamental and applied studies at the Engineering Cybernetics department (headed by him), at Systems Studies department at the Institute of Cybernetics of AS UkrSSR, and earlier at Kyiv Institute of Civil Aircraft Engineers (KIIGA), where he was a chairman and a chief science officer. The principles of consistency and systems analysis assume that an object, process or a problem are considered and analyzed as a system taking into account all its sides and their interrelations. If we represent system sides as manifolds and their interrelations as mappings between them, then we obtain an appropriate mathematical model of this system. We also consider problems on models. The concept of a model and problems on a model were implicitly used by designers and developers of Mir-2 system and Analitik language, who worked at the Institute of Cybernetics AS UkrSSR under the 408 1060-0396/16/5203-0408 © 2016 Springer Science+Business Media New York 1 V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, aik@public.icyb.kiev.ua. 2 National Aviation University, Kyiv, Ukraine, † kharch@nau.edu.ua; ‡ glanm@yahoo.com. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2016, pp. 83–96. Original article submitted January 18, 2016. DOI 10.1007/s10559-016-9841-2